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.mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox vcard"><caption class="infobox-title fn org">Star number</caption><tbody><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"><a href="/wiki/File:Star_number_73_as_sum_of_gnomons.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Star_number_73_as_sum_of_gnomons.svg/200px-Star_number_73_as_sum_of_gnomons.svg.png" decoding="async" width="200" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Star_number_73_as_sum_of_gnomons.svg/300px-Star_number_73_as_sum_of_gnomons.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Star_number_73_as_sum_of_gnomons.svg/400px-Star_number_73_as_sum_of_gnomons.svg.png 2x" data-file-width="296" data-file-height="338" /></a></span><div class="infobox-caption">First four star numbers, by color.</div></td></tr><tr><th scope="row" class="infobox-label">Total <abbr title="number">no.</abbr> of terms</th><td class="infobox-data"><a href="/wiki/Infinity" title="Infinity">infinity</a></td></tr><tr><th scope="row" class="infobox-label">Formula</th><td class="infobox-data"><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 S_{n}=6n(n-1)+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=6n(n-1)+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/850d51469b3ffae5c3b1729edef223b77d624fd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.509ex; height:2.843ex;" alt="{\displaystyle S_{n}=6n(n-1)+1}"></span></td></tr><tr><th scope="row" class="infobox-label">First terms</th><td class="infobox-data"><a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">1</a>, <a href="/wiki/13_(number)" title="13 (number)">13</a>, <a href="/wiki/37_(number)" title="37 (number)">37</a>, <a href="/wiki/73_(number)" title="73 (number)">73</a>, <a href="/wiki/121_(number)" title="121 (number)">121</a>, <a href="/wiki/181_(number)" title="181 (number)">181</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> index</th><td class="infobox-data"><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="plainlist"><ul><li><a rel="nofollow" class="external text" href="//oeis.org/A003154">A003154</a></li><li>star</li></ul></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Chinese_checkers_start.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Chinese_checkers_start.svg/220px-Chinese_checkers_start.svg.png" decoding="async" width="220" height="245" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Chinese_checkers_start.svg/330px-Chinese_checkers_start.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Chinese_checkers_start.svg/440px-Chinese_checkers_start.svg.png 2x" data-file-width="620" data-file-height="691" /></a><figcaption>The Chinese checkers board has 121 holes.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>star number</b> is a <a href="/wiki/Centered_number" class="mw-redirect" title="Centered number">centered</a> <a href="/wiki/Figurate_number" title="Figurate number">figurate number</a>, a centered <a href="/wiki/Hexagram" title="Hexagram">hexagram</a> (six-pointed star), such as the Star of David, or the board <a href="/wiki/Chinese_checkers" title="Chinese checkers">Chinese checkers</a> is played on. </p> <table style="line-height: 1em; min-width: 290px;" cellspacing="10"> <tbody><tr> <th>1</th> <th></th> <th>13</th> <th></th> <th>37 </th></tr> <tr align="center" valign="middle"> <td><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" 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class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" 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decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> </td></tr></tbody></table> <p>The <i>n</i>th star number is given by the formula <i>S<sub>n</sub></i> = 6<i>n</i>(<i>n</i> − 1) + 1. The first 45 star numbers are 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837, 11353, and 11881. (sequence <span class="nowrap external"><a href="//oeis.org/A003154" class="extiw" title="oeis:A003154">A003154</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </p><p>The <a href="/wiki/Digital_root" title="Digital root">digital root</a> of a star number is always 1 or 4, and progresses in the sequence 1, 4, 1. The last two digits of a star number in base 10 are always 01, 13, 21, 33, 37, 41, 53, 61, 73, 81, or 93. </p><p>Unique among the star numbers is 35113, since its prime factors (i.e., 13, 37 and 73) are also consecutive star numbers. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Relationships_to_other_kinds_of_numbers">Relationships to other kinds of numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Star_number&action=edit&section=1" title="Edit section: Relationships to other kinds of numbers"><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:Star_number_visual_proof.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Star_number_visual_proof.svg/220px-Star_number_visual_proof.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Star_number_visual_proof.svg/330px-Star_number_visual_proof.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Star_number_visual_proof.svg/440px-Star_number_visual_proof.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption><a href="/wiki/Proof_without_words" title="Proof without words">Proof without words</a> that the <i>n</i>-th star number is 12 times the (<i>n</i>−1)-th triangular number, plus one</figcaption></figure> <p>Geometrically, the <i>n</i>th star number is made up of a central point and 12 copies of the (<i>n</i>−1)th <a href="/wiki/Triangular_number" title="Triangular number">triangular number</a> — making it numerically equal to the <i>n</i>th <a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered dodecagonal number</a>, but differently arranged. As such, the formula the <i>n</i>th star number can be written as S_n=1+12T_n-1 where T_n=n(n+1)/2. </p><p>Infinitely many star numbers are also <a href="/wiki/Triangular_number" title="Triangular number">triangular numbers</a>, the first four being <i>S</i><sub>1</sub> = 1 = <i>T</i><sub>1</sub>, <i>S</i><sub>7</sub> = 253 = <i>T</i><sub>22</sub>, <i>S</i><sub>91</sub> = 49141 = <i>T</i><sub>313</sub>, and <i>S</i><sub>1261</sub> = 9533161 = <i>T</i><sub>4366</sub> (sequence <span class="nowrap external"><a href="//oeis.org/A156712" class="extiw" title="oeis:A156712">A156712</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p><p>Infinitely many star numbers are also <a href="/wiki/Square_number" title="Square number">square numbers</a>, the first four being <i>S</i><sub>1</sub> = 1<sup>2</sup>, <i>S</i><sub>5</sub> = 121 = 11<sup>2</sup>, <i>S</i><sub>45</sub> = 11881 = 109<sup>2</sup>, and <i>S</i><sub>441</sub> = 1164241 = 1079<sup>2</sup> (sequence <span class="nowrap external"><a href="//oeis.org/A054318" class="extiw" title="oeis:A054318">A054318</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), for square stars (sequence <span class="nowrap external"><a href="//oeis.org/A006061" class="extiw" title="oeis:A006061">A006061</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p><p>A <b>star prime</b> is a star number that is <a href="/wiki/Prime_number" title="Prime number">prime</a>. The first few star primes (sequence <span class="nowrap external"><a href="//oeis.org/A083577" class="extiw" title="oeis:A083577">A083577</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) are 13, 37, 73, 181, 337, 433, 541, 661, 937. </p><p>A <b>superstar prime</b> is a star prime whose prime index is also a star number. The first two such numbers are 661 and 1750255921. </p><p>A <b>reverse superstar prime</b> is a star number whose index is a star prime. The first few such numbers are 937, 7993, 31537, 195481, 679393, 1122337, 1752841, 2617561, 5262193. </p><p>The term "star number" or "stellate number" is occasionally used to refer to <a href="/wiki/Octagonal_number" title="Octagonal number">octagonal numbers</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> </p> <div class="mw-heading mw-heading2"><h2 id="Other_properties">Other properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Star_number&action=edit&section=2" title="Edit section: Other properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">harmonic series</a> of <a href="/wiki/Unit_fraction" title="Unit fraction">unit fractions</a> with the star numbers as denominators 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 {\begin{aligned}\sum _{n=1}^{\infty }&{\frac {1}{S_{n}}}\\&=1+{\frac {1}{13}}+{\frac {1}{37}}+{\frac {1}{73}}+{\frac {1}{121}}+{\frac {1}{181}}+{\frac {1}{253}}+{\frac {1}{337}}+\cdots \\&={\frac {\pi }{2{\sqrt {3}}}}\tan({\frac {\pi }{2{\sqrt {3}}}})\\&\approx 1.159173.\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>13</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>37</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>73</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>121</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>181</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>253</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>337</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mn>1.159173.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }&{\frac {1}{S_{n}}}\\&=1+{\frac {1}{13}}+{\frac {1}{37}}+{\frac {1}{73}}+{\frac {1}{121}}+{\frac {1}{181}}+{\frac {1}{253}}+{\frac {1}{337}}+\cdots \\&={\frac {\pi }{2{\sqrt {3}}}}\tan({\frac {\pi }{2{\sqrt {3}}}})\\&\approx 1.159173.\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de92f62566dd6e408b695065199a0a7fc499f995" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:60.591ex; height:21.176ex;" alt="{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }&{\frac {1}{S_{n}}}\\&=1+{\frac {1}{13}}+{\frac {1}{37}}+{\frac {1}{73}}+{\frac {1}{121}}+{\frac {1}{181}}+{\frac {1}{253}}+{\frac {1}{337}}+\cdots \\&={\frac {\pi }{2{\sqrt {3}}}}\tan({\frac {\pi }{2{\sqrt {3}}}})\\&\approx 1.159173.\\\end{aligned}}}"></span> </p><p>The <a href="/wiki/Alternating_series" title="Alternating series">alternating series</a> of <a href="/wiki/Unit_fraction" title="Unit fraction">unit fractions</a> with the star numbers as denominators 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 {\begin{aligned}\sum _{n=1}^{\infty }&(-1)^{n-1}{\frac {1}{S_{n}}}\\&=1-{\frac {1}{13}}+{\frac {1}{37}}-{\frac {1}{73}}+{\frac {1}{121}}-{\frac {1}{181}}+{\frac {1}{253}}-{\frac {1}{337}}+\cdots \\&\approx 0.941419.\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> </mtd> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>13</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>37</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>73</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>121</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>181</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>253</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>337</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mn>0.941419.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }&(-1)^{n-1}{\frac {1}{S_{n}}}\\&=1-{\frac {1}{13}}+{\frac {1}{37}}-{\frac {1}{73}}+{\frac {1}{121}}-{\frac {1}{181}}+{\frac {1}{253}}-{\frac {1}{337}}+\cdots \\&\approx 0.941419.\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f316fe14781ca37fd468ca6d62c61b0fbdbe6b33" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:60.591ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }&(-1)^{n-1}{\frac {1}{S_{n}}}\\&=1-{\frac {1}{13}}+{\frac {1}{37}}-{\frac {1}{73}}+{\frac {1}{121}}-{\frac {1}{181}}+{\frac {1}{253}}-{\frac {1}{337}}+\cdots \\&\approx 0.941419.\\\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Star_number&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal number</a></li></ul> <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=Star_number&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><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"><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="CITEREFSloane_"A000567"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A000567">"Sequence A000567"</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%3BA000567&rft_id=https%3A%2F%2Foeis.org%2FA000567&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStar+number" class="Z3988"></span></span> </li> </ol></div> <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 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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 numbers</a></li> <li><a href="/wiki/Square_number" title="Square number">Square numbers</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal numbers</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal numbers</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal numbers</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal numbers</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal numbers</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal numbers</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal numbers</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 hlist" 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 numbers</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube numbers</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral numbers</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral numbers</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral numbers</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/Cube_(algebra)" title="Cube (algebra)">Cube numbers</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral numbers</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral numbers</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral numbers</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula numbers</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/Tetrahedral_number" title="Tetrahedral number">Tetrahedral numbers</a></li> <li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal numbers</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 hlist" 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 numbers</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular numbers</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic numbers</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Higher <a href="/wiki/Dimension" title="Dimension">dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="non-centered12" scope="row" class="navbox-group" style="width:1%">non-centered</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/Fifth_power_(algebra)" title="Fifth power (algebra)">5-hypercube numbers</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">6-hypercube numbers</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">7-hypercube numbers</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">8-hypercube numbers</a></li></ul> </div></td></tr></tbody></table><div></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_numbers743" 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_numbers743" 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_numbers743" 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_±_1743" 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_numbers743" 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_numbers743" 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_numbers743" 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_sums743" 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_numbers743" 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 class="mw-selflink selflink">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_numbers743" 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="Primes743" 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="Pseudoprimes743" 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_dynamics743" 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_numbers743" 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 href="/wiki/Regular_number" title="Regular number">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_numbers743" 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_numbers743" 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_sieve743" 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_related743" 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_related743" 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" 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