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<span class="vector-toc-numb">2</span> <span>Babylonian mathematics</span> </div> </a> <ul id="toc-Babylonian_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Music_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Music_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Music theory</span> </div> </a> <ul id="toc-Music_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algorithms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Algorithms</span> </div> </a> <ul id="toc-Algorithms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_applications"> <div 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-good-star" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="/wiki/Wikipedia:Good_articles*" title="This is a good article. Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p class="mw-empty-elt"> </p> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Numbers that evenly divide powers of 60</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Regular_prime" title="Regular prime">regular prime</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Regular_divisibility_lattice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Regular_divisibility_lattice.svg/300px-Regular_divisibility_lattice.svg.png" decoding="async" width="300" height="178" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Regular_divisibility_lattice.svg/450px-Regular_divisibility_lattice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Regular_divisibility_lattice.svg/600px-Regular_divisibility_lattice.svg.png 2x" data-file-width="1363" data-file-height="809" /></a><figcaption>A <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> of <a href="/wiki/Divisibility" class="mw-redirect" title="Divisibility">divisibility</a> relationships among the regular numbers up to 400. The vertical scale is <a href="/wiki/Logarithmic_scale" title="Logarithmic scale">logarithmic</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p><b>Regular numbers</b> are numbers that evenly divide powers of <a href="/wiki/60_(number)" title="60 (number)">60</a> (or, equivalently, powers of <a href="/wiki/30_(number)" title="30 (number)">30</a>). Equivalently, they are the numbers whose only prime divisors are <a href="/wiki/2" title="2">2</a>, <a href="/wiki/3" title="3">3</a>, and <a href="/wiki/5" title="5">5</a>. As an example, 60<sup>2</sup> = 3600 = 48 × 75, so as divisors of a power of 60 both 48 and 75 are regular. </p><p>These numbers arise in several areas of mathematics and its applications, and have different names coming from their different areas of study. </p> <ul><li>In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, these numbers are called <b>5-smooth</b>, because they can be characterized as having only 2, 3, or 5 as their <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factors</a>. This is a specific case of the more general <span class="texhtml mvar" style="font-style:italic;">k</span>-<a href="/wiki/Smooth_number" title="Smooth number">smooth numbers</a>, the numbers that have no prime factor greater <span class="nowrap">than <span class="texhtml mvar" style="font-style:italic;">k</span>.</span></li> <li>In the study of <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylonian mathematics</a>, the divisors of powers of 60 are called <b>regular numbers</b> or <b>regular sexagesimal numbers</b>, and are of great importance in this area because of the <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> (base 60) number system that the Babylonians used for writing their numbers, and that was central to Babylonian mathematics.</li> <li>In <a href="/wiki/Music_theory" title="Music theory">music theory</a>, regular numbers occur in the ratios of tones in <a href="/wiki/Five-limit_tuning" title="Five-limit tuning">five-limit</a> <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a>. In connection with music theory and related theories of <a href="/wiki/Architecture" title="Architecture">architecture</a>, these numbers have been called the <b>harmonic whole numbers</b>.</li> <li>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, regular numbers are often called <b>Hamming numbers</b>, after <a href="/wiki/Richard_Hamming" title="Richard Hamming">Richard Hamming</a>, who proposed the problem of finding computer <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> for generating these numbers in ascending order. This problem has been used as a test case for <a href="/wiki/Functional_programming" title="Functional programming">functional programming</a>.</li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Number_theory">Number theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_number&action=edit&section=1" title="Edit section: Number theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formally, a regular number is an <a href="/wiki/Integer" title="Integer">integer</a> of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{i}\cdot 3^{j}\cdot 5^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{i}\cdot 3^{j}\cdot 5^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a58f3dd6fd24933574a5079a3c5ec60354b8e95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.644ex; height:2.676ex;" alt="{\displaystyle 2^{i}\cdot 3^{j}\cdot 5^{k}}"></span>, for nonnegative integers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>. Such a number is a divisor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 60^{\max(\lceil i\,/2\rceil ,j,k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⌈</mo> <mi>i</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌉</mo> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60^{\max(\lceil i\,/2\rceil ,j,k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c415f906653ade5768c6f14480cf7a79b820b7eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.403ex; height:2.843ex;" alt="{\displaystyle 60^{\max(\lceil i\,/2\rceil ,j,k)}}"></span>. The regular numbers are also called 5-<a href="/wiki/Smooth_number" title="Smooth number">smooth</a>, indicating that their greatest <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> is at most 5.<sup id="cite_ref-FOOTNOTESloane_"A051037"_2-0" class="reference"><a href="#cite_note-FOOTNOTESloane_"A051037"-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> More generally, a <span class="texhtml mvar" style="font-style:italic;">k</span>-smooth number is a number whose greatest prime factor is at <span class="nowrap">most <span class="texhtml mvar" style="font-style:italic;">k</span>.<sup id="cite_ref-FOOTNOTEPomerance1995_3-0" class="reference"><a href="#cite_note-FOOTNOTEPomerance1995-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></span> </p><p>The first few regular numbers are<sup id="cite_ref-FOOTNOTESloane_"A051037"_2-1" class="reference"><a href="#cite_note-FOOTNOTESloane_"A051037"-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.6em;">1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, ... (sequence <span class="nowrap external"><a href="//oeis.org/A051037" class="extiw" title="oeis:A051037">A051037</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</div> <p>Several other sequences at the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a> have definitions involving 5-smooth numbers.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Although the regular numbers appear dense within the range from 1 to 60, they are quite sparse among the larger integers. A regular number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2^{i}\cdot 3^{j}\cdot 5^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2^{i}\cdot 3^{j}\cdot 5^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d19d3b0ea9876eecf3ac3e5060cd93c46b71b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.137ex; height:2.676ex;" alt="{\displaystyle n=2^{i}\cdot 3^{j}\cdot 5^{k}}"></span> is less than or equal to some threshold <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> if and only if the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca15aed635d1550057e6ea4cc1a5d125d225706f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.849ex; height:2.843ex;" alt="{\displaystyle (i,j,k)}"></span> belongs to the <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a> bounded by the coordinate planes and the plane <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\ln 2+j\ln 3+k\ln 5\leq \ln N,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <mo>+</mo> <mi>j</mi> <mi>ln</mi> <mo>⁡</mo> <mn>3</mn> <mo>+</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mn>5</mn> <mo>≤</mo> <mi>ln</mi> <mo>⁡</mo> <mi>N</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\ln 2+j\ln 3+k\ln 5\leq \ln N,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e032ba86783597b5e801bbc4ec08bf364a7db1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.416ex; height:2.509ex;" alt="{\displaystyle i\ln 2+j\ln 3+k\ln 5\leq \ln N,}"></span> as can be seen by taking logarithms of both sides of the inequality <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{i}\cdot 3^{j}\cdot 5^{k}\leq N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>≤</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{i}\cdot 3^{j}\cdot 5^{k}\leq N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a8200ce9e7449f57e2075fa92e7d9d5f43a6c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.806ex; height:2.843ex;" alt="{\displaystyle 2^{i}\cdot 3^{j}\cdot 5^{k}\leq N}"></span>. Therefore, the number of regular numbers that are at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> can be estimated as the <a href="/wiki/Volume" title="Volume">volume</a> of this tetrahedron, which is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\log _{2}N\,\log _{3}N\,\log _{5}N}{6}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>⁡</mo> <mi>N</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\log _{2}N\,\log _{3}N\,\log _{5}N}{6}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa54047b16d85bc71d38e3ec3ca48b261fb788af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.461ex; height:5.509ex;" alt="{\displaystyle {\frac {\log _{2}N\,\log _{3}N\,\log _{5}N}{6}}.}"></span> Even more precisely, using <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a>, the number of regular numbers up to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\left(\ln(N{\sqrt {30}})\right)^{3}}{6\ln 2\ln 3\ln 5}}+O(\log N),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>30</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>6</mn> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <mi>ln</mi> <mo>⁡</mo> <mn>3</mn> <mi>ln</mi> <mo>⁡</mo> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left(\ln(N{\sqrt {30}})\right)^{3}}{6\ln 2\ln 3\ln 5}}+O(\log N),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4caebc5e2096ba87a550c361fe9a64dcf4ff2f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.586ex; height:6.843ex;" alt="{\displaystyle {\frac {\left(\ln(N{\sqrt {30}})\right)^{3}}{6\ln 2\ln 3\ln 5}}+O(\log N),}"></span> and it has been conjectured that the error term of this approximation is actually <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\log \log N)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log \log N)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2f6c9502c8747f6316e77d77d1d582e4262f65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.364ex; height:2.843ex;" alt="{\displaystyle O(\log \log N)}"></span>.<sup id="cite_ref-FOOTNOTESloane_"A051037"_2-2" class="reference"><a href="#cite_note-FOOTNOTESloane_"A051037"-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A similar formula for the number of 3-smooth numbers up to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is given by <a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a> in his first letter to <a href="/wiki/G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a>.<sup id="cite_ref-FOOTNOTEBerndtRankin1995_5-0" class="reference"><a href="#cite_note-FOOTNOTEBerndtRankin1995-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Babylonian_mathematics">Babylonian mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_number&action=edit&section=2" title="Edit section: Babylonian mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Table_de_division_et_de_conversion_des_fractions_-_Louvre_-_AO_6456.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Table_de_division_et_de_conversion_des_fractions_-_Louvre_-_AO_6456.JPG/300px-Table_de_division_et_de_conversion_des_fractions_-_Louvre_-_AO_6456.JPG" decoding="async" width="300" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Table_de_division_et_de_conversion_des_fractions_-_Louvre_-_AO_6456.JPG/450px-Table_de_division_et_de_conversion_des_fractions_-_Louvre_-_AO_6456.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Table_de_division_et_de_conversion_des_fractions_-_Louvre_-_AO_6456.JPG/600px-Table_de_division_et_de_conversion_des_fractions_-_Louvre_-_AO_6456.JPG 2x" data-file-width="5184" data-file-height="3456" /></a><figcaption>AO 6456, a table of reciprocals of regular numbers from <a href="/wiki/Uruk#Uruk_into_Late_Antiquity" title="Uruk">Seleucid Uruk</a>, copied from an unknown earlier source</figcaption></figure> <p>In the Babylonian <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> notation, the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> of a regular number has a finite representation. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> divides <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 60^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce69788b14a9fc51f544bd0c8d52f62a0bf5d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.414ex; height:2.676ex;" alt="{\displaystyle 60^{k}}"></span>, then the sexagesimal representation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e10667bad240500f5044257143510127e03d69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.72ex; height:2.843ex;" alt="{\displaystyle 1/n}"></span> is just that for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 60^{k}/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60^{k}/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3676d6bc306fb93f669583ced7a536cc0da64eda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.971ex; height:3.176ex;" alt="{\displaystyle 60^{k}/n}"></span>, shifted by some number of places. This allows for easy division by these numbers: to divide by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, multiply by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e10667bad240500f5044257143510127e03d69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.72ex; height:2.843ex;" alt="{\displaystyle 1/n}"></span>, then shift.<sup id="cite_ref-FOOTNOTEAaboe1965_6-0" class="reference"><a href="#cite_note-FOOTNOTEAaboe1965-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>For instance, consider division by the regular number 54 = 2<sup>1</sup>3<sup>3</sup>. 54 is a divisor of 60<sup>3</sup>, and 60<sup>3</sup>/54 = 4000, so dividing by 54 in sexagesimal can be accomplished by multiplying by 4000 and shifting three places. In sexagesimal 4000 = 1×3600 + 6×60 + 40×1, or (as listed by Joyce) 1:6:40. Thus, 1/54, in sexagesimal, is 1/60 + 6/60<sup>2</sup> + 40/60<sup>3</sup>, also denoted 1:6:40 as Babylonian notational conventions did not specify the power of the starting digit. Conversely 1/4000 = 54/60<sup>3</sup>, so division by 1:6:40 = 4000 can be accomplished by instead multiplying by 54 and shifting three sexagesimal places. </p><p>The Babylonians used tables of reciprocals of regular numbers, some of which still survive.<sup id="cite_ref-FOOTNOTESachs1947_7-0" class="reference"><a href="#cite_note-FOOTNOTESachs1947-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> These tables existed relatively unchanged throughout Babylonian times.<sup id="cite_ref-FOOTNOTEAaboe1965_6-1" class="reference"><a href="#cite_note-FOOTNOTEAaboe1965-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> One tablet from <a href="/wiki/Seleucid" class="mw-redirect" title="Seleucid">Seleucid</a> times, by someone named Inaqibıt-Anu, contains the reciprocals of 136 of the 231 six-place regular numbers whose first place is 1 or 2, listed in order. It also includes reciprocals of some numbers of more than six places, such as 3<sup>23</sup> (2 1 4 8 3 0 7 in sexagesimal), whose reciprocal has 17 sexagesimal digits. Noting the difficulty of both calculating these numbers and sorting them, <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a> in 1972 hailed Inaqibıt-Anu as "the first man in history to solve a computational problem that takes longer than one second of time on a modern electronic computer!" (Two tables are also known giving approximations of reciprocals of non-regular numbers, one of which gives reciprocals for all the numbers from 56 to 80.)<sup id="cite_ref-FOOTNOTEKnuth1972_8-0" class="reference"><a href="#cite_note-FOOTNOTEKnuth1972-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEFowlerRobson1998_9-0" class="reference"><a href="#cite_note-FOOTNOTEFowlerRobson1998-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Although the primary reason for preferring regular numbers to other numbers involves the finiteness of their reciprocals, some Babylonian calculations other than reciprocals also involved regular numbers. For instance, tables of regular squares have been found<sup id="cite_ref-FOOTNOTEAaboe1965_6-2" class="reference"><a href="#cite_note-FOOTNOTEAaboe1965-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and the broken tablet <a href="/wiki/Plimpton_322" title="Plimpton 322">Plimpton 322</a> has been interpreted by <a href="/wiki/Otto_E._Neugebauer" title="Otto E. Neugebauer">Neugebauer</a> as listing <a href="/wiki/Pythagorean_triple" title="Pythagorean triple">Pythagorean triples</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mi>p</mi> <mi>q</mi> <mo>,</mo> <mspace width="thinmathspace" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe74e431048405e9c9204497d839aa2709e0d6b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.448ex; height:3.176ex;" alt="{\displaystyle (p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})}"></span> generated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> both regular and less than 60.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Fowler and Robson discuss the calculation of square roots, such as how the Babylonians found an approximation to the <a href="/wiki/Square_root_of_2" title="Square root of 2">square root of 2</a>, perhaps using regular number approximations of fractions such as 17/12.<sup id="cite_ref-FOOTNOTEFowlerRobson1998_9-1" class="reference"><a href="#cite_note-FOOTNOTEFowlerRobson1998-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Music_theory">Music theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_number&action=edit&section=3" title="Edit section: Music theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Music_theory" title="Music theory">music theory</a>, the <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a> of the <a href="/wiki/Diatonic_scale" title="Diatonic scale">diatonic scale</a> involves regular numbers: the <a href="/wiki/Pitch_(music)" title="Pitch (music)">pitches</a> in a single <a href="/wiki/Octave" title="Octave">octave</a> of this scale have frequencies proportional to the numbers in the sequence 24, 27, 30, 32, 36, 40, 45, 48 of nearly consecutive regular numbers.<sup id="cite_ref-FOOTNOTEClarke1877_11-0" class="reference"><a href="#cite_note-FOOTNOTEClarke1877-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Thus, for an instrument with this tuning, all pitches are regular-number <a href="/wiki/Harmonic" title="Harmonic">harmonics</a> of a single <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a>. This scale is called a 5-<a href="/wiki/Limit_(music)" title="Limit (music)">limit</a> tuning, meaning that the <a href="/wiki/Interval_(music)" title="Interval (music)">interval</a> between any two pitches can be described as a product 2<sup>i</sup>3<sup>j</sup>5<sup>k</sup> of powers of the prime numbers up to 5, or equivalently as a ratio of regular numbers.<sup id="cite_ref-FOOTNOTEHoninghBod2005_12-0" class="reference"><a href="#cite_note-FOOTNOTEHoninghBod2005-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>5-limit musical scales other than the familiar diatonic scale of Western music have also been used, both in traditional musics of other cultures and in modern experimental music: <a href="#CITEREFHoninghBod2005">Honingh & Bod (2005)</a> list 31 different 5-limit scales, drawn from a larger database of musical scales. Each of these 31 scales shares with diatonic just intonation the property that all intervals are ratios of regular numbers.<sup id="cite_ref-FOOTNOTEHoninghBod2005_12-1" class="reference"><a href="#cite_note-FOOTNOTEHoninghBod2005-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a>'s <a href="/wiki/Tonnetz" title="Tonnetz">tonnetz</a> provides a convenient graphical representation of the pitches in any 5-limit tuning, by factoring out the octave relationships (powers of two) so that the remaining values form a planar <a href="/wiki/Regular_grid" title="Regular grid">grid</a>.<sup id="cite_ref-FOOTNOTEHoninghBod2005_12-2" class="reference"><a href="#cite_note-FOOTNOTEHoninghBod2005-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Some music theorists have stated more generally that regular numbers are fundamental to tonal music itself, and that pitch ratios based on primes larger than 5 cannot be <a href="/wiki/Consonance_and_dissonance" title="Consonance and dissonance">consonant</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> However the <a href="/wiki/Equal_temperament" title="Equal temperament">equal temperament</a> of modern pianos is not a 5-limit tuning,<sup id="cite_ref-FOOTNOTEKopiez2003_14-0" class="reference"><a href="#cite_note-FOOTNOTEKopiez2003-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> and some modern composers have experimented with tunings based on primes larger than five.<sup id="cite_ref-FOOTNOTEWolf2003_15-0" class="reference"><a href="#cite_note-FOOTNOTEWolf2003-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>In connection with the application of regular numbers to music theory, it is of interest to find pairs of regular numbers that differ by one. There are exactly ten such pairs <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,x+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,x+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21080e22a669e76fc222f9a857547d332e603ddd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.505ex; height:2.843ex;" alt="{\displaystyle (x,x+1)}"></span> and each such pair defines a <a href="/wiki/Superparticular_number" class="mw-redirect" title="Superparticular number">superparticular ratio</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {x+1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>x</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {x+1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adb74d89104dd97468c8ab4f29be464ad80bb34f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.877ex; height:3.509ex;" alt="{\displaystyle {\tfrac {x+1}{x}}}"></span> that is meaningful as a musical interval. These intervals are 2/1 (the <a href="/wiki/Octave" title="Octave">octave</a>), 3/2 (the <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a>), 4/3 (the <a href="/wiki/Perfect_fourth" title="Perfect fourth">perfect fourth</a>), 5/4 (the <a href="/wiki/Just_major_third" class="mw-redirect" title="Just major third">just major third</a>), 6/5 (the <a href="/wiki/Just_minor_third" class="mw-redirect" title="Just minor third">just minor third</a>), 9/8 (the <a href="/wiki/Just_major_tone" class="mw-redirect" title="Just major tone">just major tone</a>), 10/9 (the <a href="/wiki/Just_minor_tone" class="mw-redirect" title="Just minor tone">just minor tone</a>), 16/15 (the <a href="/wiki/Just_diatonic_semitone" class="mw-redirect" title="Just diatonic semitone">just diatonic semitone</a>), 25/24 (the <a href="/wiki/Just_chromatic_semitone" class="mw-redirect" title="Just chromatic semitone">just chromatic semitone</a>), and 81/80 (the <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a>).<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>In the Renaissance theory of <a href="/wiki/Musica_universalis" title="Musica universalis">universal harmony</a>, musical ratios were used in other applications, including the <a href="/wiki/Architecture" title="Architecture">architecture</a> of buildings. In connection with the analysis of these shared musical and architectural ratios, for instance in the architecture of <a href="/wiki/Palladio" class="mw-redirect" title="Palladio">Palladio</a>, the regular numbers have also been called the <b>harmonic whole numbers</b>.<sup id="cite_ref-FOOTNOTEHowardLongair1982_17-0" class="reference"><a href="#cite_note-FOOTNOTEHowardLongair1982-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Algorithms">Algorithms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_number&action=edit&section=4" title="Edit section: Algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Wikifunctions-logo.svg/40px-Wikifunctions-logo.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Wikifunctions-logo.svg/60px-Wikifunctions-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Wikifunctions-logo.svg/80px-Wikifunctions-logo.svg.png 2x" data-file-width="512" data-file-height="513" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikifunctions" title="Wikifunctions">Wikifunctions</a> has <b><a href="https://www.wikifunctions.org/wiki/Z15224" class="extiw" title="f:Z15224">a regular number checking function</a></b>.</div></div> </div> <p>Algorithms for calculating the regular numbers in ascending order were popularized by <a href="/wiki/Edsger_Dijkstra" class="mw-redirect" title="Edsger Dijkstra">Edsger Dijkstra</a>. Dijkstra (<a href="#CITEREFDijkstra1976">1976</a>, <a href="#CITEREFDijkstra1981">1981</a>) attributes to Hamming the problem of building the infinite ascending sequence of all 5-smooth numbers; this problem is now known as <b>Hamming's problem</b>, and the numbers so generated are also called the <b>Hamming numbers</b>. Dijkstra's ideas to compute these numbers are the following: </p> <ul><li>The sequence of Hamming numbers begins with the number 1.</li> <li>The remaining values in the sequence are of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961075004bd1ffd53a4685965e9eb3a9b691f645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.501ex; height:2.176ex;" alt="{\displaystyle 2h}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d25f14284625d3bc8a551e4cf0b7700f00eb8ae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.501ex; height:2.176ex;" alt="{\displaystyle 3h}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ce1a5e4064b64b01d4ef49ae811be5bfc72313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.501ex; height:2.176ex;" alt="{\displaystyle 5h}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> is any Hamming number.</li> <li>Therefore, the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> may be generated by outputting the value 1, and then <a href="/wiki/Merge_algorithm" title="Merge algorithm">merging</a> the sequences <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d8809b4c2d94baa01903caafdb15d47b900232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.226ex; height:2.176ex;" alt="{\displaystyle 2H}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/097477e4725a01f954a5fce4b30ce08e1d135b17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.226ex; height:2.176ex;" alt="{\displaystyle 3H}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fef27e1b344a7342c2984c6b2199e27ea8b66df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.226ex; height:2.176ex;" alt="{\displaystyle 5H}"></span>.</li></ul> <p>This algorithm is often used to demonstrate the power of a <a href="/wiki/Lazy_evaluation" title="Lazy evaluation">lazy</a> <a href="/wiki/Functional_programming_language" class="mw-redirect" title="Functional programming language">functional programming language</a>, because (implicitly) concurrent efficient implementations, using a constant number of arithmetic operations per generated value, are easily constructed as described above. Similarly efficient strict functional or <a href="/wiki/Imperative_programming_language" class="mw-redirect" title="Imperative programming language">imperative</a> sequential implementations are also possible whereas explicitly concurrent <a href="/wiki/Generator_(computer_programming)" title="Generator (computer programming)">generative</a> solutions might be non-trivial.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>In the <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python programming language</a>, lazy functional code for generating regular numbers is used as one of the built-in tests for correctness of the language's implementation.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>A related problem, discussed by <a href="#CITEREFKnuth1972">Knuth (1972)</a>, is to list all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>-digit sexagesimal numbers in ascending order (see <a href="#Babylonian_mathematics">#Babylonian mathematics</a> above). In algorithmic terms, this is equivalent to generating (in order) the subsequence of the infinite sequence of regular numbers, ranging from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 60^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce69788b14a9fc51f544bd0c8d52f62a0bf5d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.414ex; height:2.676ex;" alt="{\displaystyle 60^{k}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 60^{k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60^{k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/052a55a1e23ddcd3753c8817339fd89c4ec253d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.514ex; height:2.676ex;" alt="{\displaystyle 60^{k+1}}"></span>.<sup id="cite_ref-FOOTNOTEKnuth1972_8-1" class="reference"><a href="#cite_note-FOOTNOTEKnuth1972-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> See <a href="#CITEREFGingerich1965">Gingerich (1965)</a> for an early description of computer code that generates these numbers out of order and then sorts them;<sup id="cite_ref-FOOTNOTEGingerich1965_20-0" class="reference"><a href="#cite_note-FOOTNOTEGingerich1965-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Knuth describes an ad hoc algorithm, which he attributes to <a href="#CITEREFBruins1970">Bruins (1970)</a>, for generating the six-digit numbers more quickly but that does not generalize in a straightforward way to larger values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>.<sup id="cite_ref-FOOTNOTEKnuth1972_8-2" class="reference"><a href="#cite_note-FOOTNOTEKnuth1972-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="#CITEREFEppstein2007">Eppstein (2007)</a> describes an algorithm for computing tables of this type in linear time for arbitrary values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>.<sup id="cite_ref-FOOTNOTEEppstein2007_21-0" class="reference"><a href="#cite_note-FOOTNOTEEppstein2007-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Other_applications">Other applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_number&action=edit&section=5" title="Edit section: Other applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="#CITEREFHeningerRainsSloane2006">Heninger, Rains & Sloane (2006)</a> show that, when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is a regular number and is divisible by 8, the generating function of an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-dimensional extremal even <a href="/wiki/Unimodular_lattice" title="Unimodular lattice">unimodular lattice</a> is an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>th power of a polynomial.<sup id="cite_ref-FOOTNOTEHeningerRainsSloane2006_22-0" class="reference"><a href="#cite_note-FOOTNOTEHeningerRainsSloane2006-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>As with other classes of <a href="/wiki/Smooth_number" title="Smooth number">smooth numbers</a>, regular numbers are important as problem sizes in computer programs for performing the <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a>, a technique for analyzing the dominant frequencies of signals in <a href="/wiki/Signal_(electrical_engineering)" class="mw-redirect" title="Signal (electrical engineering)">time-varying data</a>. For instance, the method of <a href="#CITEREFTemperton1992">Temperton (1992)</a> requires that the transform length be a regular number.<sup id="cite_ref-FOOTNOTETemperton1992_23-0" class="reference"><a href="#cite_note-FOOTNOTETemperton1992-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>Book VIII of <a href="/wiki/Plato" title="Plato">Plato</a>'s <i><a href="/wiki/The_Republic_(Plato)" class="mw-redirect" title="The Republic (Plato)">Republic</a></i> involves an allegory of marriage centered on the highly regular number 60<sup>4</sup> = 12,960,000 and its divisors (see <a href="/wiki/Plato%27s_number" title="Plato's number">Plato's number</a>). Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>Certain species of <a href="/wiki/Bamboo" title="Bamboo">bamboo</a> release large numbers of seeds in synchrony (a process called <a href="/wiki/Mast_(botany)" class="mw-redirect" title="Mast (botany)">masting</a>) at intervals that have been estimated as regular numbers of years, with different intervals for different species, including examples with intervals of 10, 15, 16, 30, 32, 48, 60, and 120 years.<sup id="cite_ref-FOOTNOTEVellerNowakDavis2015_25-0" class="reference"><a href="#cite_note-FOOTNOTEVellerNowakDavis2015-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> It has been hypothesized that the biological mechanism for timing and synchronizing this process lends itself to smooth numbers, and in particular in this case to 5-smooth numbers. Although the estimated masting intervals for some other species of bamboo are not regular numbers of years, this may be explainable as measurement error.<sup id="cite_ref-FOOTNOTEVellerNowakDavis2015_25-1" class="reference"><a href="#cite_note-FOOTNOTEVellerNowakDavis2015-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_number&action=edit&section=6" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Inspired by similar diagrams by Erkki Kurenniemi in <a rel="nofollow" class="external text" href="http://www.beige.org/projects/dimi/CSDL2.pdf">"Chords, scales, and divisor lattices"</a>.</span> </li> <li id="cite_note-FOOTNOTESloane_"A051037"-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTESloane_"A051037"_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTESloane_"A051037"_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTESloane_"A051037"_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFSloane_"A051037"">Sloane "A051037"</a>.</span> </li> <li id="cite_note-FOOTNOTEPomerance1995-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPomerance1995_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPomerance1995">Pomerance (1995)</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.oeis.org/search?q=5-smooth">OEIS search for sequences involving 5-smoothness</a>.</span> </li> <li id="cite_note-FOOTNOTEBerndtRankin1995-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBerndtRankin1995_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBerndtRankin1995">Berndt & Rankin (1995)</a>.</span> </li> <li id="cite_note-FOOTNOTEAaboe1965-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEAaboe1965_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAaboe1965_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAaboe1965_6-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFAaboe1965">Aaboe (1965)</a>.</span> </li> <li id="cite_note-FOOTNOTESachs1947-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESachs1947_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSachs1947">Sachs (1947)</a>.</span> </li> <li id="cite_note-FOOTNOTEKnuth1972-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEKnuth1972_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEKnuth1972_8-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEKnuth1972_8-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFKnuth1972">Knuth (1972)</a>.</span> </li> <li id="cite_note-FOOTNOTEFowlerRobson1998-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEFowlerRobson1998_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEFowlerRobson1998_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFFowlerRobson1998">Fowler & Robson (1998)</a>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">See <a href="#CITEREFConwayGuy1996">Conway & Guy (1996)</a> for a popular treatment of this interpretation. <a href="/wiki/Plimpton_322" title="Plimpton 322">Plimpton 322</a> has other interpretations, for which see its article, but all involve regular numbers.</span> </li> <li id="cite_note-FOOTNOTEClarke1877-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEClarke1877_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFClarke1877">Clarke (1877)</a>.</span> </li> <li id="cite_note-FOOTNOTEHoninghBod2005-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEHoninghBod2005_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEHoninghBod2005_12-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEHoninghBod2005_12-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFHoninghBod2005">Honingh & Bod (2005)</a>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFAsmussen2001">Asmussen (2001)</a>, for instance, states that "within any piece of tonal music" all intervals must be ratios of regular numbers, echoing similar statements by much earlier writers such as <a href="#CITEREFHabens1889">Habens (1889)</a>. In the modern music theory literature this assertion is often attributed to <a href="#CITEREFLonguet-Higgins1962">Longuet-Higgins (1962)</a>, who used a graphical arrangement closely related to the <a href="/wiki/Tonnetz" title="Tonnetz">tonnetz</a> to organize 5-limit pitches.</span> </li> <li id="cite_note-FOOTNOTEKopiez2003-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKopiez2003_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKopiez2003">Kopiez (2003)</a>.</span> </li> <li id="cite_note-FOOTNOTEWolf2003-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWolf2003_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWolf2003">Wolf (2003)</a>.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFHalseyHewitt1972">Halsey & Hewitt (1972)</a> note that this follows from <a href="/wiki/St%C3%B8rmer%27s_theorem" title="Størmer's theorem">Størmer's theorem</a> (<a href="#CITEREFStørmer1897">Størmer 1897</a>), and provide a proof for this case; see also <a href="#CITEREFSilver1971">Silver (1971)</a>.</span> </li> <li id="cite_note-FOOTNOTEHowardLongair1982-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHowardLongair1982_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHowardLongair1982">Howard & Longair (1982)</a>.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">See, e.g., <a href="#CITEREFHemmendinger1988">Hemmendinger (1988)</a> or <a href="#CITEREFYuen1992">Yuen (1992)</a>.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Function m235 in <a rel="nofollow" class="external text" href="http://svn.python.org/projects/python/trunk/Lib/test/test_generators.py">test_generators.py</a>.</span> </li> <li id="cite_note-FOOTNOTEGingerich1965-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGingerich1965_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGingerich1965">Gingerich (1965)</a>.</span> </li> <li id="cite_note-FOOTNOTEEppstein2007-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEppstein2007_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEppstein2007">Eppstein (2007)</a>.</span> </li> <li id="cite_note-FOOTNOTEHeningerRainsSloane2006-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHeningerRainsSloane2006_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHeningerRainsSloane2006">Heninger, Rains & Sloane (2006)</a>.</span> </li> <li id="cite_note-FOOTNOTETemperton1992-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETemperton1992_23-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTemperton1992">Temperton (1992)</a>.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><a href="#CITEREFBarton1908">Barton (1908)</a>; <a href="#CITEREFMcClain1974">McClain (1974)</a>.</span> </li> <li id="cite_note-FOOTNOTEVellerNowakDavis2015-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEVellerNowakDavis2015_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEVellerNowakDavis2015_25-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFVellerNowakDavis2015">Veller, Nowak & Davis (2015)</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_number&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAaboe1965" class="citation cs2"><a href="/wiki/Aaboe" class="mw-redirect" title="Aaboe">Aaboe, Asger</a> (1965), "Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)", <i><a href="/wiki/Journal_of_Cuneiform_Studies" title="Journal of Cuneiform Studies">Journal of Cuneiform Studies</a></i>, <b>19</b> (3), The American Schools of Oriental Research: 79–86, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1359089">10.2307/1359089</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1359089">1359089</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0191779">0191779</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:164195082">164195082</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Cuneiform+Studies&rft.atitle=Some+Seleucid+mathematical+tables+%28extended+reciprocals+and+squares+of+regular+numbers%29&rft.volume=19&rft.issue=3&rft.pages=79-86&rft.date=1965&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A164195082%23id-name%3DS2CID&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0191779%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1359089%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1359089&rft.aulast=Aaboe&rft.aufirst=Asger&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAsmussen2001" class="citation cs2">Asmussen, Robert (2001), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160424195738/http://www.terraworld.net/c-jasmussen/thesis_asmussen.pdf"><i>Periodicity of sinusoidal frequencies as a basis for the analysis of Baroque and Classical harmony: a computer based study</i></a> <span class="cs1-format">(PDF)</span>, Ph.D. thesis, <a href="/wiki/University_of_Leeds" title="University of Leeds">University of Leeds</a>, archived from <a rel="nofollow" class="external text" href="http://www.terraworld.net/c-jasmussen/thesis_asmussen.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2016-04-24<span class="reference-accessdate">, retrieved <span class="nowrap">2007-03-15</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Periodicity+of+sinusoidal+frequencies+as+a+basis+for+the+analysis+of+Baroque+and+Classical+harmony%3A+a+computer+based+study&rft.series=Ph.D.+thesis&rft.pub=University+of+Leeds&rft.date=2001&rft.aulast=Asmussen&rft.aufirst=Robert&rft_id=http%3A%2F%2Fwww.terraworld.net%2Fc-jasmussen%2Fthesis_asmussen.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarton1908" class="citation cs2">Barton, George A. (1908), "On the Babylonian origin of Plato's nuptial number", <i><a href="/wiki/Journal_of_the_American_Oriental_Society" title="Journal of the American Oriental Society">Journal of the American Oriental Society</a></i>, <b>29</b>, American Oriental Society: 210–219, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F592627">10.2307/592627</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/592627">592627</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Oriental+Society&rft.atitle=On+the+Babylonian+origin+of+Plato%27s+nuptial+number&rft.volume=29&rft.pages=210-219&rft.date=1908&rft_id=info%3Adoi%2F10.2307%2F592627&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F592627%23id-name%3DJSTOR&rft.aulast=Barton&rft.aufirst=George+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndtRankin1995" class="citation cs2">Berndt, Bruce C.; Rankin, Robert Alexander, eds. (1995), <i>Ramanujan: letters and commentary</i>, History of mathematics, vol. 9, American Mathematical Society, p. 23, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995rlc..book.....B">1995rlc..book.....B</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0470-4" title="Special:BookSources/978-0-8218-0470-4"><bdi>978-0-8218-0470-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%3A+letters+and+commentary&rft.series=History+of+mathematics&rft.pages=23&rft.pub=American+Mathematical+Society&rft.date=1995&rft_id=info%3Abibcode%2F1995rlc..book.....B&rft.isbn=978-0-8218-0470-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBruins1970" class="citation cs2">Bruins, E. M. (1970), "La construction de la grande table le valeurs réciproques AO 6456", in Finet, André (ed.), <i>Actes de la XVII<sup>e</sup> Rencontre Assyriologique Internationale</i>, Comité belge de recherches en Mésopotamie, pp. 99–115</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=La+construction+de+la+grande+table+le+valeurs+r%C3%A9ciproques+AO+6456&rft.btitle=Actes+de+la+XVII%3Csup%3Ee%3C%2Fsup%3E+Rencontre+Assyriologique+Internationale&rft.pages=99-115&rft.pub=Comit%C3%A9+belge+de+recherches+en+M%C3%A9sopotamie&rft.date=1970&rft.aulast=Bruins&rft.aufirst=E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClarke1877" class="citation cs2">Clarke, A. R. (January 1877), "Just intonation", <i>Nature</i>, <b>15</b> (377): 253, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1877Natur..15..253C">1877Natur..15..253C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F015253b0">10.1038/015253b0</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Just+intonation&rft.volume=15&rft.issue=377&rft.pages=253&rft.date=1877-01&rft_id=info%3Adoi%2F10.1038%2F015253b0&rft_id=info%3Abibcode%2F1877Natur..15..253C&rft.aulast=Clarke&rft.aufirst=A.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConwayGuy1996" class="citation cs2"><a href="/wiki/John_H._Conway" class="mw-redirect" title="John H. Conway">Conway, John H.</a>; <a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (1996), <a rel="nofollow" class="external text" href="https://archive.org/details/bookofnumbers0000conw/page/172"><i>The Book of Numbers</i></a>, Copernicus, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/bookofnumbers0000conw/page/172">172–176</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97993-X" title="Special:BookSources/0-387-97993-X"><bdi>0-387-97993-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Book+of+Numbers&rft.pages=172-176&rft.pub=Copernicus&rft.date=1996&rft.isbn=0-387-97993-X&rft.aulast=Conway&rft.aufirst=John+H.&rft.au=Guy%2C+Richard+K.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbookofnumbers0000conw%2Fpage%2F172&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDijkstra1976" class="citation cs2"><a href="/wiki/Edsger_W._Dijkstra" title="Edsger W. Dijkstra">Dijkstra, Edsger W.</a> (1976), "17. An exercise attributed to R. W. Hamming", <a rel="nofollow" class="external text" href="https://archive.org/details/disciplineofprog0000dijk/page/129"><i>A Discipline of Programming</i></a>, Prentice-Hall, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/disciplineofprog0000dijk/page/129">129–134</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0132158718" title="Special:BookSources/978-0132158718"><bdi>978-0132158718</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=17.+An+exercise+attributed+to+R.+W.+Hamming&rft.btitle=A+Discipline+of+Programming&rft.pages=129-134&rft.pub=Prentice-Hall&rft.date=1976&rft.isbn=978-0132158718&rft.aulast=Dijkstra&rft.aufirst=Edsger+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdisciplineofprog0000dijk%2Fpage%2F129&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDijkstra1981" class="citation cs2"><a href="/wiki/Edsger_W._Dijkstra" title="Edsger W. Dijkstra">Dijkstra, Edsger W.</a> (1981), <a rel="nofollow" class="external text" href="http://www.cs.utexas.edu/users/EWD/ewd07xx/EWD792.PDF"><i>Hamming's exercise in SASL</i></a> <span class="cs1-format">(PDF)</span>, Report EWD792. Originally a privately circulated handwritten note</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hamming%27s+exercise+in+SASL&rft.date=1981&rft.aulast=Dijkstra&rft.aufirst=Edsger+W.&rft_id=http%3A%2F%2Fwww.cs.utexas.edu%2Fusers%2FEWD%2Fewd07xx%2FEWD792.PDF&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEppstein2007" class="citation cs2"><a href="/wiki/David_Eppstein" title="David Eppstein">Eppstein, David</a> (2007), <a rel="nofollow" class="external text" href="https://11011110.github.io/blog/2007/03/18/range-restricted-hamming-problem.html"><i>The range-restricted Hamming problem</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+range-restricted+Hamming+problem&rft.date=2007&rft.aulast=Eppstein&rft.aufirst=David&rft_id=https%3A%2F%2F11011110.github.io%2Fblog%2F2007%2F03%2F18%2Frange-restricted-hamming-problem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFowlerRobson1998" class="citation cs2">Fowler, David; Robson, Eleanor (1998), <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0315086098922091/pdf?md5=7c3e116fac2deef21b107b78f3561862&pid=1-s2.0-S0315086098922091-main.pdf">"Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context"</a> <span class="cs1-format">(PDF)</span>, <i>Historia Mathematica</i>, <b>25</b> (4): 366–378, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fhmat.1998.2209">10.1006/hmat.1998.2209</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=Square+Root+Approximations+in+Old+Babylonian+Mathematics%3A+YBC+7289+in+Context&rft.volume=25&rft.issue=4&rft.pages=366-378&rft.date=1998&rft_id=info%3Adoi%2F10.1006%2Fhmat.1998.2209&rft.aulast=Fowler&rft.aufirst=David&rft.au=Robson%2C+Eleanor&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS0315086098922091%2Fpdf%3Fmd5%3D7c3e116fac2deef21b107b78f3561862%26pid%3D1-s2.0-S0315086098922091-main.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>, page 375.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGingerich1965" class="citation cs2">Gingerich, Owen (1965), "Eleven-digit regular sexagesimals and their reciprocals", <i><a href="/wiki/Transactions_of_the_American_Philosophical_Society" class="mw-redirect" title="Transactions of the American Philosophical Society">Transactions of the American Philosophical Society</a></i>, <b>55</b> (8), American Philosophical Society: 3–38, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1006080">10.2307/1006080</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1006080">1006080</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Philosophical+Society&rft.atitle=Eleven-digit+regular+sexagesimals+and+their+reciprocals&rft.volume=55&rft.issue=8&rft.pages=3-38&rft.date=1965&rft_id=info%3Adoi%2F10.2307%2F1006080&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1006080%23id-name%3DJSTOR&rft.aulast=Gingerich&rft.aufirst=Owen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHabens1889" class="citation cs2">Habens, Rev. W. J. (1889), <a rel="nofollow" class="external text" href="https://zenodo.org/record/2404277">"On the musical scale"</a>, <i>Proceedings of the Musical Association</i>, <b>16</b>, Royal Musical Association: 16th Session, p. 1, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/765355">765355</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Musical+Association&rft.atitle=On+the+musical+scale&rft.volume=16&rft.pages=16th+Session%2C+p.+1&rft.date=1889&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F765355%23id-name%3DJSTOR&rft.aulast=Habens&rft.aufirst=Rev.+W.+J.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F2404277&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalseyHewitt1972" class="citation cs2">Halsey, G. D.; Hewitt, Edwin (1972), "More on the superparticular ratios in music", <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>, <b>79</b> (10), Mathematical Association of America: 1096–1100, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2317424">10.2307/2317424</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2317424">2317424</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0313189">0313189</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=More+on+the+superparticular+ratios+in+music&rft.volume=79&rft.issue=10&rft.pages=1096-1100&rft.date=1972&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0313189%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2317424%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2317424&rft.aulast=Halsey&rft.aufirst=G.+D.&rft.au=Hewitt%2C+Edwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHemmendinger1988" class="citation cs2">Hemmendinger, David (1988), "The "Hamming problem" in Prolog", <i><a href="/wiki/ACM_SIGPLAN_Notices" class="mw-redirect" title="ACM SIGPLAN Notices">ACM SIGPLAN Notices</a></i>, <b>23</b> (4): 81–86, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F44326.44335">10.1145/44326.44335</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:28906392">28906392</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+SIGPLAN+Notices&rft.atitle=The+%22Hamming+problem%22+in+Prolog&rft.volume=23&rft.issue=4&rft.pages=81-86&rft.date=1988&rft_id=info%3Adoi%2F10.1145%2F44326.44335&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A28906392%23id-name%3DS2CID&rft.aulast=Hemmendinger&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeningerRainsSloane2006" class="citation cs2"><a href="/wiki/Nadia_Heninger" title="Nadia Heninger">Heninger, Nadia</a>; Rains, E. M.; <a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (2006), "On the integrality of <span class="texhtml mvar" style="font-style:italic;">n</span>th roots of generating functions", <i><a href="/wiki/Journal_of_Combinatorial_Theory" title="Journal of Combinatorial Theory">Journal of Combinatorial Theory</a></i>, Series A, <b>113</b> (8): 1732–1745, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.NT/0509316">math.NT/0509316</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jcta.2006.03.018">10.1016/j.jcta.2006.03.018</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2269551">2269551</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15913795">15913795</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Combinatorial+Theory&rft.atitle=On+the+integrality+of+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3En%3C%2Fspan%3Eth+roots+of+generating+functions&rft.volume=113&rft.issue=8&rft.pages=1732-1745&rft.date=2006&rft_id=info%3Aarxiv%2Fmath.NT%2F0509316&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2269551%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15913795%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.jcta.2006.03.018&rft.aulast=Heninger&rft.aufirst=Nadia&rft.au=Rains%2C+E.+M.&rft.au=Sloane%2C+N.+J.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>}.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoninghBod2005" class="citation cs2">Honingh, Aline; Bod, Rens (2005), "Convexity and the well-formedness of musical objects", <i><a href="/wiki/Journal_of_New_Music_Research" title="Journal of New Music Research">Journal of New Music Research</a></i>, <b>34</b> (3): 293–303, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F09298210500280612">10.1080/09298210500280612</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16321292">16321292</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+New+Music+Research&rft.atitle=Convexity+and+the+well-formedness+of+musical+objects&rft.volume=34&rft.issue=3&rft.pages=293-303&rft.date=2005&rft_id=info%3Adoi%2F10.1080%2F09298210500280612&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16321292%23id-name%3DS2CID&rft.aulast=Honingh&rft.aufirst=Aline&rft.au=Bod%2C+Rens&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHowardLongair1982" class="citation cs2"><a href="/wiki/Deborah_Howard" title="Deborah Howard">Howard, Deborah</a>; Longair, Malcolm (May 1982), "Harmonic proportion and Palladio's <i>Quattro Libri</i>", <i>Journal of the Society of Architectural Historians</i>, <b>41</b> (2): 116–143, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F989675">10.2307/989675</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/989675">989675</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Society+of+Architectural+Historians&rft.atitle=Harmonic+proportion+and+Palladio%27s+Quattro+Libri&rft.volume=41&rft.issue=2&rft.pages=116-143&rft.date=1982-05&rft_id=info%3Adoi%2F10.2307%2F989675&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F989675%23id-name%3DJSTOR&rft.aulast=Howard&rft.aufirst=Deborah&rft.au=Longair%2C+Malcolm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1972" class="citation cs2"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, D. E.</a> (1972), <a rel="nofollow" class="external text" href="https://teaching.csse.uwa.edu.au/units/CITS1001/extension/ancient-babylonian-algorithms.pdf">"Ancient Babylonian algorithms"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a></i>, <b>15</b> (7): 671–677, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F361454.361514">10.1145/361454.361514</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7829945">7829945</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=Ancient+Babylonian+algorithms&rft.volume=15&rft.issue=7&rft.pages=671-677&rft.date=1972&rft_id=info%3Adoi%2F10.1145%2F361454.361514&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7829945%23id-name%3DS2CID&rft.aulast=Knuth&rft.aufirst=D.+E.&rft_id=https%3A%2F%2Fteaching.csse.uwa.edu.au%2Funits%2FCITS1001%2Fextension%2Fancient-babylonian-algorithms.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>. A correction appears in <a rel="nofollow" class="external text" href="https://dl.acm.org/doi/pdf/10.1145/359997.360014"><i>CACM</i> <b>19</b>(2), 1976</a>, stating that the tablet does not contain all 231 of the numbers of interest. The article (corrected) with a brief addendum appears in <a rel="nofollow" class="external text" href="https://doc.lagout.org/science/0_Computer%20Science/7_Technical%20Papers/Selected%20Papers%20on%20Computer%20Science%20-%20Donald%20E.%20Knuth.pdf"><i>Selected Papers on Computer Science</i>, CSLI Lecture Notes 59, Cambridge Univ. Press, 1996, pp. 185–203</a>, but without the Appendix that was included in the original paper.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKopiez2003" class="citation cs2">Kopiez, Reinhard (2003), "Intonation of harmonic intervals: adaptability of expert musicians to equal temperament and just intonation", <i>Music Perception</i>, <b>20</b> (4): 383–410, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1525%2Fmp.2003.20.4.383">10.1525/mp.2003.20.4.383</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Music+Perception&rft.atitle=Intonation+of+harmonic+intervals%3A+adaptability+of+expert+musicians+to+equal+temperament+and+just+intonation&rft.volume=20&rft.issue=4&rft.pages=383-410&rft.date=2003&rft_id=info%3Adoi%2F10.1525%2Fmp.2003.20.4.383&rft.aulast=Kopiez&rft.aufirst=Reinhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLonguet-Higgins1962" class="citation cs2"><a href="/wiki/H._Christopher_Longuet-Higgins" class="mw-redirect" title="H. Christopher Longuet-Higgins">Longuet-Higgins, H. C.</a> (1962), "Letter to a musical friend", <i>Music Review</i> (August): 244–248</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Music+Review&rft.atitle=Letter+to+a+musical+friend&rft.issue=August&rft.pages=244-248&rft.date=1962&rft.aulast=Longuet-Higgins&rft.aufirst=H.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcClain1974" class="citation cs2"><a href="/wiki/Ernest_G._McClain" title="Ernest G. McClain">McClain, Ernest G.</a> (1974), "Musical "Marriages" in Plato's "Republic"<span class="cs1-kern-right"></span>", <i><a href="/wiki/Journal_of_Music_Theory" title="Journal of Music Theory">Journal of Music Theory</a></i>, <b>18</b> (2), Duke University Press: 242–272, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/843638">843638</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Music+Theory&rft.atitle=Musical+%22Marriages%22+in+Plato%27s+%22Republic%22&rft.volume=18&rft.issue=2&rft.pages=242-272&rft.date=1974&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F843638%23id-name%3DJSTOR&rft.aulast=McClain&rft.aufirst=Ernest+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPomerance1995" class="citation cs2"><a href="/wiki/Carl_Pomerance" title="Carl Pomerance">Pomerance, Carl</a> (1995), "The role of smooth numbers in number-theoretic algorithms", <i>Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994)</i>, Basel: Birkhäuser, pp. 411–422, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1403941">1403941</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+role+of+smooth+numbers+in+number-theoretic+algorithms&rft.btitle=Proceedings+of+the+International+Congress+of+Mathematicians%2C+Vol.+1%2C+2+%28Z%C3%BCrich%2C+1994%29&rft.place=Basel&rft.pages=411-422&rft.pub=Birkh%C3%A4user&rft.date=1995&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1403941%23id-name%3DMR&rft.aulast=Pomerance&rft.aufirst=Carl&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSachs1947" class="citation cs2">Sachs, A. J. (1947), "Babylonian mathematical texts. I. Reciprocals of regular sexagesimal numbers", <i><a href="/wiki/Journal_of_Cuneiform_Studies" title="Journal of Cuneiform Studies">Journal of Cuneiform Studies</a></i>, <b>1</b> (3), The American Schools of Oriental Research: 219–240, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1359434">10.2307/1359434</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1359434">1359434</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0022180">0022180</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:163783242">163783242</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Cuneiform+Studies&rft.atitle=Babylonian+mathematical+texts.+I.+Reciprocals+of+regular+sexagesimal+numbers&rft.volume=1&rft.issue=3&rft.pages=219-240&rft.date=1947&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A163783242%23id-name%3DS2CID&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0022180%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1359434%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1359434&rft.aulast=Sachs&rft.aufirst=A.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilver1971" class="citation cs2">Silver, A. L. Leigh (1971), "Musimatics or the nun's fiddle", <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>, <b>78</b> (4), Mathematical Association of America: 351–357, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2316896">10.2307/2316896</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2316896">2316896</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Musimatics+or+the+nun%27s+fiddle&rft.volume=78&rft.issue=4&rft.pages=351-357&rft.date=1971&rft_id=info%3Adoi%2F10.2307%2F2316896&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2316896%23id-name%3DJSTOR&rft.aulast=Silver&rft.aufirst=A.+L.+Leigh&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane_"A051037"" class="citation web cs2"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.), <a rel="nofollow" class="external text" href="https://oeis.org/A051037">"Sequence A051037 (5-smooth numbers)"</a>, <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>, OEIS Foundation</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA051037%26%23x20%3B%285-smooth+numbers%29&rft_id=https%3A%2F%2Foeis.org%2FA051037&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStørmer1897" class="citation cs2"><a href="/wiki/Carl_St%C3%B8rmer" title="Carl Størmer">Størmer, Carl</a> (1897), "Quelques théorèmes sur l'équation de Pell <span class="texhtml"><i>x</i><sup>2</sup> − <i>Dy</i><sup>2</sup> = ±1</span> et leurs applications", <i>Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl.</i>, <b>I</b> (2)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Skrifter+Videnskabs-selskabet+%28Christiania%29%2C+Mat.-Naturv.+Kl.&rft.atitle=Quelques+th%C3%A9or%C3%A8mes+sur+l%27%C3%A9quation+de+Pell+%3Cspan+class%3D%22texhtml+%22+%3Ex%3Csup%3E2%3C%2Fsup%3E+%26minus%3B+Dy%3Csup%3E2%3C%2Fsup%3E+%3D+%C2%B11%3C%2Fspan%3E+et+leurs+applications&rft.volume=I&rft.issue=2&rft.date=1897&rft.aulast=St%C3%B8rmer&rft.aufirst=Carl&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTemperton1992" class="citation cs2">Temperton, Clive (1992), "A generalized prime factor FFT algorithm for any <span class="texhtml"><i>N</i> = 2<sup><i>p</i></sup>3<sup><i>q</i></sup>5<sup><i>r</i></sup></span>", <i>SIAM Journal on Scientific and Statistical Computing</i>, <b>13</b> (3): 676–686, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0913039">10.1137/0913039</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14764894">14764894</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Scientific+and+Statistical+Computing&rft.atitle=A+generalized+prime+factor+FFT+algorithm+for+any+%3Cspan+class%3D%22texhtml+%22+%3EN+%3D+2%3Csup%3Ep%3C%2Fsup%3E3%3Csup%3Eq%3C%2Fsup%3E5%3Csup%3Er%3C%2Fsup%3E%3C%2Fspan%3E&rft.volume=13&rft.issue=3&rft.pages=676-686&rft.date=1992&rft_id=info%3Adoi%2F10.1137%2F0913039&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14764894%23id-name%3DS2CID&rft.aulast=Temperton&rft.aufirst=Clive&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVellerNowakDavis2015" class="citation cs2">Veller, Carl; Nowak, Martin A.; Davis, Charles C. (May 2015), "Extended flowering intervals of bamboos evolved by discrete multiplication", <i>Ecology Letters</i>, <b>18</b> (7): 653–659, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015EcolL..18..653V">2015EcolL..18..653V</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fele.12442">10.1111/ele.12442</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25963600">25963600</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ecology+Letters&rft.atitle=Extended+flowering+intervals+of+bamboos+evolved+by+discrete+multiplication&rft.volume=18&rft.issue=7&rft.pages=653-659&rft.date=2015-05&rft_id=info%3Apmid%2F25963600&rft_id=info%3Adoi%2F10.1111%2Fele.12442&rft_id=info%3Abibcode%2F2015EcolL..18..653V&rft.aulast=Veller&rft.aufirst=Carl&rft.au=Nowak%2C+Martin+A.&rft.au=Davis%2C+Charles+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolf2003" class="citation cs2">Wolf, Daniel James (March 2003), "Alternative tunings, alternative tonalities", <i>Contemporary Music Review</i>, <b>22</b> (1–2): 3–14, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0749446032000134715">10.1080/0749446032000134715</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:191457676">191457676</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Contemporary+Music+Review&rft.atitle=Alternative+tunings%2C+alternative+tonalities&rft.volume=22&rft.issue=1%E2%80%932&rft.pages=3-14&rft.date=2003-03&rft_id=info%3Adoi%2F10.1080%2F0749446032000134715&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A191457676%23id-name%3DS2CID&rft.aulast=Wolf&rft.aufirst=Daniel+James&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYuen1992" class="citation cs2">Yuen, C. K. (1992), "Hamming numbers, lazy evaluation, and eager disposal", <i><a href="/wiki/ACM_SIGPLAN_Notices" class="mw-redirect" title="ACM SIGPLAN Notices">ACM SIGPLAN Notices</a></i>, <b>27</b> (8): 71–75, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F142137.142151">10.1145/142137.142151</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18283005">18283005</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+SIGPLAN+Notices&rft.atitle=Hamming+numbers%2C+lazy+evaluation%2C+and+eager+disposal&rft.volume=27&rft.issue=8&rft.pages=71-75&rft.date=1992&rft_id=info%3Adoi%2F10.1145%2F142137.142151&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18283005%23id-name%3DS2CID&rft.aulast=Yuen&rft.aufirst=C.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARegular+number" class="Z3988"></span>.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Regular_number&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/ma105/reciprocals3600.html">Table of reciprocals of regular numbers up to 3600</a> from the web site of Professor David E. Joyce, Clark University.</li> <li><a rel="nofollow" class="external text" href="http://rosettacode.org/wiki/Hamming_numbers">RosettaCode</a> Generation of Hamming_numbers in ~ 50 programming languages</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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id="Divisibility-based_sets_of_integers" style="font-size:114%;margin:0 4em">Divisibility-based sets of integers</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integer_factorization" title="Integer factorization">Integer factorization</a></li> <li><a href="/wiki/Divisor" title="Divisor">Divisor</a></li> <li><a href="/wiki/Unitary_divisor" title="Unitary divisor">Unitary divisor</a></li> <li><a href="/wiki/Divisor_function" title="Divisor function">Divisor function</a></li> <li><a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">Prime factor</a></li> <li><a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">Fundamental theorem of arithmetic</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Lattice_of_the_divisibility_of_60.svg" class="mw-file-description" title="Divisibility of 60"><img alt="Divisibility of 60" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/175px-Lattice_of_the_divisibility_of_60.svg.png" decoding="async" width="175" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/263px-Lattice_of_the_divisibility_of_60.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/350px-Lattice_of_the_divisibility_of_60.svg.png 2x" data-file-width="313" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Factorization forms</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prime_number" title="Prime number">Prime</a></li> <li><a href="/wiki/Composite_number" title="Composite number">Composite</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/Square-free_integer" title="Square-free integer">Square-free</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a class="mw-selflink selflink">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Unusual_number" title="Unusual number">Unusual</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constrained divisor sums</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li> <li><a href="/wiki/Unitary_perfect_number" title="Unitary perfect number">Unitary perfect</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With many divisors</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Weird_number" title="Weird number">Weird</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequence</a>-related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li> <li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a> (<a href="/wiki/Amicable_triple" title="Amicable triple">Triple</a>)</li> <li><a href="/wiki/Sociable_number" title="Sociable number">Sociable</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other sets</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Friendly_number#Solitary_numbers" title="Friendly number">Solitary</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Classes_of_natural_numbers" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classes_of_natural_numbers" title="Template:Classes of natural numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classes_of_natural_numbers" title="Template talk:Classes of natural numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classes_of_natural_numbers" title="Special:EditPage/Template:Classes of natural numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Classes_of_natural_numbers" style="font-size:114%;margin:0 4em">Classes of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Exponentiation" title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of 2</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Power of 3</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Power of 10</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cube</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Fourth power</a></li> <li><a href="/wiki/Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">Sixth power</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">Seventh power</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">Eighth power</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Prime_power" title="Prime power">Prime power</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cullen_number" title="Cullen number">Cullen</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li> <li><a href="/wiki/Fermat_number" title="Fermat number">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell</a></li> <li><a href="/wiki/Perrin_number" title="Perrin number">Perrin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_number" title="Amenable number">Amenable</a></li> <li><a href="/wiki/Congruent_number" title="Congruent number">Congruent</a></li> <li><a href="/wiki/Kn%C3%B6del_number" title="Knödel number">Knödel</a></li> <li><a href="/wiki/Riesel_number" title="Riesel number">Riesel</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li> <li><a href="/wiki/Polite_number" title="Polite number">Polite</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li> <li><a href="/wiki/Ulam_number" title="Ulam number">Ulam</a></li> <li><a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li> <li><a href="/wiki/Centered_square_number" title="Centered square number">Centered square</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li> <li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li> <li><a href="/wiki/Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li> <li><a href="/wiki/Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li> <li><a href="/wiki/Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li> <li><a href="/wiki/Star_number" title="Star number">Star</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triangular_number" title="Triangular number">Triangular</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Square_triangular_number" title="Square triangular number">Square triangular</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentatope_number" title="Pentatope number">Pentatope</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_number" title="Bell number">Bell</a></li> <li><a href="/wiki/Cake_number" title="Cake number">Cake</a></li> <li><a href="/wiki/Catalan_number" title="Catalan number">Catalan</a></li> <li><a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind</a></li> <li><a href="/wiki/Delannoy_number" title="Delannoy number">Delannoy</a></li> <li><a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler</a></li> <li><a href="/wiki/Eulerian_number" title="Eulerian number">Eulerian</a></li> <li><a href="/wiki/Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li> <li><a href="/wiki/Lah_number" title="Lah number">Lah</a></li> <li><a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li> <li><a href="/wiki/Lobb_number" title="Lobb number">Lobb</a></li> <li><a href="/wiki/Motzkin_number" title="Motzkin number">Motzkin</a></li> <li><a href="/wiki/Narayana_number" title="Narayana number">Narayana</a></li> <li><a href="/wiki/Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li> <li><a href="/wiki/Schr%C3%B6der_number" title="Schröder number">Schröder</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li> <li><a href="/wiki/Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li> <li><a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li> <li><a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li> <li><a href="/wiki/Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li> <li><a href="/wiki/Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes" style="font-size:114%;margin:0 4em"><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li> <li><a href="/wiki/Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics" style="font-size:114%;margin:0 4em"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Highly_totient_number" title="Highly totient number">Highly totient</a></li> <li><a href="/wiki/Noncototient" title="Noncototient">Noncototient</a></li> <li><a href="/wiki/Nontotient" title="Nontotient">Nontotient</a></li> <li><a href="/wiki/Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li> <li><a href="/wiki/Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li> <li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclid_number" title="Euclid number">Euclid</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers" style="font-size:114%;margin:0 4em">Other <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="/wiki/Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blum_integer" title="Blum integer">Blum</a></li> <li><a href="/wiki/Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Giuga_number" title="Giuga number">Giuga</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a class="mw-selflink selflink">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/St%C3%B8rmer_number" title="Størmer number">Størmer</a></li> <li><a href="/wiki/Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> <br />and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Persistence_of_a_number" title="Persistence of a number">Persistence</a> <ul><li><a href="/wiki/Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li> <li><a href="/wiki/Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></li> <li><a href="/wiki/Digital_root" title="Digital root">Digital root</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Meertens_number" title="Meertens number">Meertens</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dudeney_number" title="Dudeney number">Dudeney</a></li> <li><a href="/wiki/Factorion" title="Factorion">Factorion</a></li> <li><a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar</a></li> <li><a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's constant</a></li> <li><a href="/wiki/Keith_number" title="Keith number">Keith</a></li> <li><a href="/wiki/Lychrel_number" title="Lychrel number">Lychrel</a></li> <li><a href="/wiki/Narcissistic_number" title="Narcissistic number">Narcissistic</a></li> <li><a href="/wiki/Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li> <li><a href="/wiki/Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a> <ul><li><a href="/wiki/Happy_number" title="Happy number">Happy</a></li></ul></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Automorphic_number" title="Automorphic number">Automorphic</a> <ul><li><a href="/wiki/Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_number" title="Palindromic number">Palindromic</a></li> <li><a href="/wiki/Pandigital_number" title="Pandigital number">Pandigital</a></li> <li><a href="/wiki/Repdigit" title="Repdigit">Repdigit</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit</a></li> <li><a href="/wiki/Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li> <li><a href="/wiki/Undulating_number" title="Undulating number">Undulating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="/wiki/Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cyclic_number" title="Cyclic number">Cyclic</a></li> <li><a href="/wiki/Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li> <li><a href="/wiki/Parasitic_number" title="Parasitic number">Parasitic</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Transposable_integer" title="Transposable integer">Transposable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li> <li><a href="/wiki/Vampire_number" title="Vampire number">Vampire</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Friedman_number" title="Friedman number">Friedman</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evil_number" title="Evil number">Evil</a></li> <li><a href="/wiki/Odious_number" title="Odious number">Odious</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve" style="font-size:114%;margin:0 4em">Generated via a <a href="/wiki/Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Generation_of_primes" title="Generation of primes">Prime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pancake_sorting" title="Pancake sorting">Pancake number</a></li> <li><a href="/wiki/Sorting_number" title="Sorting number">Sorting number</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aronson%27s_sequence" title="Aronson's sequence">Aronson's sequence</a></li> <li><a href="/wiki/Ban_number" title="Ban number">Ban</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a 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[\"CITEREFKnuth1972\"] = 1,\n [\"CITEREFKopiez2003\"] = 1,\n [\"CITEREFLonguet-Higgins1962\"] = 1,\n [\"CITEREFMcClain1974\"] = 1,\n [\"CITEREFPomerance1995\"] = 1,\n [\"CITEREFSachs1947\"] = 1,\n [\"CITEREFSilver1971\"] = 1,\n [\"CITEREFSloane_\u0026quot;A051037\u0026quot;\"] = 1,\n [\"CITEREFStørmer1897\"] = 1,\n [\"CITEREFTemperton1992\"] = 1,\n [\"CITEREFVellerNowakDavis2015\"] = 1,\n [\"CITEREFWolf2003\"] = 1,\n [\"CITEREFYuen1992\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Block indent\"] = 1,\n [\"Citation\"] = 30,\n [\"Cite OEIS\"] = 1,\n [\"Classes of natural numbers\"] = 1,\n [\"Distinguish\"] = 1,\n [\"Divisor classes\"] = 1,\n [\"Good article\"] = 1,\n [\"Harv\"] = 1,\n [\"Harvid\"] = 1,\n [\"Harvs\"] = 1,\n [\"Harvtxt\"] = 17,\n [\"Math\"] = 2,\n [\"Mvar\"] = 5,\n [\"Nowrap\"] = 2,\n [\"OEIS\"] = 1,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Reflist\"] = 1,\n [\"Sfnp\"] = 27,\n [\"Short description\"] = 1,\n [\"Sup\"] = 1,\n [\"Wikifunctions\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-dg26b","timestamp":"20241124054112","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Regular number","url":"https:\/\/en.wikipedia.org\/wiki\/Regular_number","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1826416","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1826416","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2006-07-13T00:36:01Z","dateModified":"2024-09-28T00:46:19Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/b\/b6\/Regular_divisibility_lattice.svg","headline":"numbers that evenly divide powers of 60"}</script> </body> </html>