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− <i>k</i> + 1</span> </div> </a> <ul id="toc-b_=_k!_−_k_+_1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Table_of_factorions_and_cycles_of_SFDb" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Table_of_factorions_and_cycles_of_SFDb"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Table of factorions and cycles of <span><span>SFD<sub><i>b</i></sub></span></span></span> </div> </a> <ul id="toc-Table_of_factorions_and_cycles_of_SFDb-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Factorion</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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dir="ltr"><p>In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, a <b>factorion</b> in a given <a href="/wiki/Number_base" class="mw-redirect" title="Number base">number base</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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> is a <a href="/wiki/Natural_number" title="Natural number">natural number</a> that equals the sum of the <a href="/wiki/Factorial" title="Factorial">factorials</a> of its <a href="/wiki/Numerical_digit" title="Numerical digit">digits</a>.<sup id="cite_ref-A014080_1-0" class="reference"><a href="#cite_note-A014080-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gardner_2-0" class="reference"><a href="#cite_note-Gardner-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Madachy_3-0" class="reference"><a href="#cite_note-Madachy-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The name factorion was coined by the author <a href="/wiki/Clifford_A._Pickover" title="Clifford A. Pickover">Clifford A. Pickover</a>.<sup id="cite_ref-Pickover_4-0" class="reference"><a href="#cite_note-Pickover-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorion&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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> be a natural number. For a base <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 b>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0041c936812fb809c4511e31eb0404de9d48511b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b>1}"></span>, we define the <b>sum of the factorials of the digits</b><sup id="cite_ref-Gupta_5-0" class="reference"><a href="#cite_note-Gupta-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-A061602_6-0" class="reference"><a href="#cite_note-A061602-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> 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 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>, <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 \operatorname {SFD} _{b}:\mathbb {N} \rightarrow \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}:\mathbb {N} \rightarrow \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6656e4f259833fd0b14f8dfe38c1576fa80df86d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.431ex; height:2.509ex;" alt="{\displaystyle \operatorname {SFD} _{b}:\mathbb {N} \rightarrow \mathbb {N} }"></span>, to be the following: </p> <dl><dd><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 \operatorname {SFD} _{b}(n)=\sum _{i=0}^{k-1}d_{i}!.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>!</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}(n)=\sum _{i=0}^{k-1}d_{i}!.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f68983ef028e71fd215ad5c6920e97a747826c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.87ex; height:7.343ex;" alt="{\displaystyle \operatorname {SFD} _{b}(n)=\sum _{i=0}^{k-1}d_{i}!.}"></span></dd></dl> <p>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 k=\lfloor \log _{b}n\rfloor +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mo fence="false" stretchy="false">⌊</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>n</mi> <mo fence="false" stretchy="false">⌋</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\lfloor \log _{b}n\rfloor +1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52db124a9d4cbd03fd312251ca6846f66ca7f673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.069ex; height:2.843ex;" alt="{\displaystyle k=\lfloor \log _{b}n\rfloor +1}"></span> is the number of digits in the number in base <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></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 n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </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/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"></span> is the <a href="/wiki/Factorial" title="Factorial">factorial</a> 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 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> and </p> <dl><dd><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 d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b^{i}}}}{b^{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <mo>−</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mrow> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b^{i}}}}{b^{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4f1f18eb0ad31fd0e63fc631ebe91c546796f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.63ex; height:5.843ex;" alt="{\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b^{i}}}}{b^{i}}}}"></span></dd></dl> <p>is the value of the <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>th digit of the number. A natural 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}"> <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 <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>-<b>factorion</b> if it is a <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a> 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 \operatorname {SFD} _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf7993d3f73b0455160ef7b3bdb28fd1c2a8a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.524ex; height:2.509ex;" alt="{\displaystyle \operatorname {SFD} _{b}}"></span>, i.e. 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 \operatorname {SFD} _{b}(n)=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}(n)=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192c83af6723fc03b79bd47308d5c67098c24cb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.221ex; height:2.843ex;" alt="{\displaystyle \operatorname {SFD} _{b}(n)=n}"></span>.<sup id="cite_ref-Abbott_7-0" class="reference"><a href="#cite_note-Abbott-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> are fixed points for all bases <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, and thus are <b>trivial factorions</b> for 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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, and all other factorions are <b>nontrivial factorions</b>. </p><p>For example, the number 145 in base <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 b=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f18909115d13fe45f000359b476549db54a104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.421ex; height:2.176ex;" alt="{\displaystyle b=10}"></span> is a factorion because <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 145=1!+4!+5!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>145</mn> <mo>=</mo> <mn>1</mn> <mo>!</mo> <mo>+</mo> <mn>4</mn> <mo>!</mo> <mo>+</mo> <mn>5</mn> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 145=1!+4!+5!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d7506f08c2a7ae9d148573b74cbfb7b081a985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.695ex; height:2.343ex;" alt="{\displaystyle 145=1!+4!+5!}"></span>. </p><p>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 b=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32584049ed5f72969777f89d69b74ee462875e82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=2}"></span>, the sum of the factorials of the digits is simply the number of digits <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> in the base 2 representation since <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 0!=1!=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0!=1!=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df097d5656bdb793a700f4c9bee417f5a009fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.978ex; height:2.176ex;" alt="{\displaystyle 0!=1!=1}"></span>. </p><p>A natural 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}"> <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 <b>sociable factorion</b> if it is a <a href="/wiki/Periodic_point" title="Periodic point">periodic point</a> 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 \operatorname {SFD} _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf7993d3f73b0455160ef7b3bdb28fd1c2a8a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.524ex; height:2.509ex;" alt="{\displaystyle \operatorname {SFD} _{b}}"></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 \operatorname {SFD} _{b}^{k}(n)=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}^{k}(n)=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046c7d6cd3ff7f0c20159e343d906071463ce6ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.372ex; height:3.176ex;" alt="{\displaystyle \operatorname {SFD} _{b}^{k}(n)=n}"></span> for a positive <a href="/wiki/Integer" title="Integer">integer</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 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>, and forms a <a href="/wiki/Periodic_sequence" title="Periodic sequence">cycle</a> of period <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>. A factorion is a sociable factorion with <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=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"></span>, and a <b>amicable factorion</b> is a sociable factorion with <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=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd301789e1f25a3da4be297ff637754ebee5f5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=2}"></span>.<sup id="cite_ref-A214285_8-0" class="reference"><a href="#cite_note-A214285-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-A254499_9-0" class="reference"><a href="#cite_note-A254499-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>All natural numbers <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> are <a href="/wiki/Periodic_point#iterated_function" title="Periodic point">preperiodic points</a> 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 \operatorname {SFD} _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf7993d3f73b0455160ef7b3bdb28fd1c2a8a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.524ex; height:2.509ex;" alt="{\displaystyle \operatorname {SFD} _{b}}"></span>, regardless of the base. This is because all natural numbers of base <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> with <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> digits satisfy <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 b^{k-1}\leq n\leq (b-1)!(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{k-1}\leq n\leq (b-1)!(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3fda822bfefaa39d3db87c00ae6859334eb274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.256ex; height:3.176ex;" alt="{\displaystyle b^{k-1}\leq n\leq (b-1)!(k)}"></span>. However, 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 k\geq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/727d7e450c0442e6585b1bf07e82d0ca14ee99d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.307ex; height:2.343ex;" alt="{\displaystyle k\geq b}"></span>, then <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 b^{k-1}>(b-1)!(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>></mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{k-1}>(b-1)!(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab1424e25b56fd93d914d3536a82733605c11c2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.762ex; height:3.176ex;" alt="{\displaystyle b^{k-1}>(b-1)!(k)}"></span> 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 b>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25519c7f5a6a77ec063bd0331afccb5b9318c439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b>2}"></span>, so any <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> will satisfy <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>\operatorname {SFD} _{b}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>\operatorname {SFD} _{b}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a03e5cfeff03c150810682fa80033afdfda12fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.221ex; height:2.843ex;" alt="{\displaystyle n>\operatorname {SFD} _{b}(n)}"></span> until <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<b^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo><</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n<b^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4293f15c9e9314b95e69f5f5411eef992170b392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.428ex; height:2.676ex;" alt="{\displaystyle n<b^{b}}"></span>. There are finitely many natural numbers less than <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 b^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4118e8754e8d89ad580f5b7629ffad805abdb68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.935ex; height:2.676ex;" alt="{\displaystyle b^{b}}"></span>, so the number is guaranteed to reach a periodic point or a fixed point less than <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 b^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4118e8754e8d89ad580f5b7629ffad805abdb68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.935ex; height:2.676ex;" alt="{\displaystyle b^{b}}"></span>, making it a preperiodic point. 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 b=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32584049ed5f72969777f89d69b74ee462875e82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=2}"></span>, the number of digits <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\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\leq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/621f658bb51d7caac329d29e9bf435361813777f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.704ex; height:2.343ex;" alt="{\displaystyle k\leq n}"></span> for any number, once again, making it a preperiodic point. This means also that there are a finite number of factorions and <a href="/wiki/Periodic_sequence" title="Periodic sequence">cycles</a> for any given base <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. </p><p>The number of iterations <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> needed 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 \operatorname {SFD} _{b}^{i}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}^{i}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5000582794ef212a3770fec53872b85c289167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.728ex; height:3.176ex;" alt="{\displaystyle \operatorname {SFD} _{b}^{i}(n)}"></span> to reach a fixed point is the <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 \operatorname {SFD} _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf7993d3f73b0455160ef7b3bdb28fd1c2a8a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.524ex; height:2.509ex;" alt="{\displaystyle \operatorname {SFD} _{b}}"></span> function's <a href="/wiki/Persistence_of_a_number" title="Persistence of a number">persistence</a> 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 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>, and undefined if it never reaches a fixed point. </p> <div class="mw-heading mw-heading2"><h2 id="Factorions_for_SFDb">Factorions for <span class="texhtml"><span class="texhtml">SFD<sub><i>b</i></sub></span></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorion&action=edit&section=2" title="Edit section: Factorions for SFDb"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="b_=_(k_−_1)!"><span id="b_.3D_.28k_.E2.88.92_1.29.21"></span><i>b</i> = (<i>k</i> − 1)!</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorion&action=edit&section=3" title="Edit section: b = (k − 1)!"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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> be a positive integer and the number base <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 b=(k-1)!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=(k-1)!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daecef719fb38746c2f9a376a77dc06e751ba56d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.766ex; height:2.843ex;" alt="{\displaystyle b=(k-1)!}"></span>. Then: </p> <ul><li><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_{1}=kb+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}=kb+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dedc2d8b2930e9ff9aaacb5ace811ddb26b2922" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.759ex; height:2.509ex;" alt="{\displaystyle n_{1}=kb+1}"></span> is a factorion 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 \operatorname {SFD} _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf7993d3f73b0455160ef7b3bdb28fd1c2a8a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.524ex; height:2.509ex;" alt="{\displaystyle \operatorname {SFD} _{b}}"></span> for 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> <mo>.</mo> </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/bcb6778a29f576eb23da1dbddffb73b2571359ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.176ex;" alt="{\displaystyle k.}"></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Proof</strong> <p>Let the digits 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 n_{1}=d_{1}b+d_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>b</mi> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}=d_{1}b+d_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79db0a62586cdc09761972dd31fc529272557487" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.912ex; height:2.509ex;" alt="{\displaystyle n_{1}=d_{1}b+d_{0}}"></span> be <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 d_{1}=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8099a0a590ada2414204f7b3648be498ec0995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.573ex; height:2.509ex;" alt="{\displaystyle d_{1}=k}"></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 d_{0}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{0}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea7ec68b7e7b94f3cabe70466cd6a2594784ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.171ex; height:2.509ex;" alt="{\displaystyle d_{0}=1.}"></span> Then </p> <dl><dd><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 \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>!</mo> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/816fb5ff46337bb0256fc10c537c3f3792a049e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.541ex; height:2.843ex;" alt="{\displaystyle \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}"></span> <dl><dd><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!+1!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>k</mi> <mo>!</mo> <mo>+</mo> <mn>1</mn> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =k!+1!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fec234de233b3fe0a44dcc930fef944c065cf5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.961ex; height:2.343ex;" alt="{\displaystyle =k!+1!}"></span></dd> <dd><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(k-1)!+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =k(k-1)!+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea688c8dab0e8aad06fdc846d4c8623b6b83102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.338ex; height:2.843ex;" alt="{\displaystyle =k(k-1)!+1}"></span></dd> <dd><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 =d_{1}b+d_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>b</mi> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =d_{1}b+d_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/967042237b16508b526e288bd1d546d78f6cd508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.818ex; height:2.509ex;" alt="{\displaystyle =d_{1}b+d_{0}}"></span></dd> <dd><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =n_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09106da232d16dab1ce5d03347e413118cdb15de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.902ex; height:2.009ex;" alt="{\displaystyle =n_{1}}"></span></dd></dl></dd></dl> <p>Thus <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee784b70e772f55ede5e6e0bdc929994bff63413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{1}}"></span> is a factorion 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 F_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b151118e130ddb3d3eea064479c152bd16738f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.432ex; height:2.509ex;" alt="{\displaystyle F_{b}}"></span> for 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>. </p> </div> <ul><li><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}=kb+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mi>b</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{2}=kb+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3595bd4419746be8d4c28ca8db5d14d0d6bb6865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.759ex; height:2.509ex;" alt="{\displaystyle n_{2}=kb+2}"></span> is a factorion 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 \operatorname {SFD} _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf7993d3f73b0455160ef7b3bdb28fd1c2a8a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.524ex; height:2.509ex;" alt="{\displaystyle \operatorname {SFD} _{b}}"></span> for 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>.</li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p>Let the digits 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 n_{2}=d_{1}b+d_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>b</mi> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{2}=d_{1}b+d_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021d4d1926c0c9dfb4c22a52901ad63676cd3e6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.912ex; height:2.509ex;" alt="{\displaystyle n_{2}=d_{1}b+d_{0}}"></span> be <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 d_{1}=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8099a0a590ada2414204f7b3648be498ec0995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.573ex; height:2.509ex;" alt="{\displaystyle d_{1}=k}"></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 d_{0}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{0}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f9169c083e8929b214b161f8d56d267b679e46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.524ex; height:2.509ex;" alt="{\displaystyle d_{0}=2}"></span>. Then </p> <dl><dd><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 \operatorname {SFD} _{b}(n_{2})=d_{1}!+d_{0}!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>!</mo> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}(n_{2})=d_{1}!+d_{0}!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a629be3bc363b80848fb819733e4bef724fe355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.541ex; height:2.843ex;" alt="{\displaystyle \operatorname {SFD} _{b}(n_{2})=d_{1}!+d_{0}!}"></span> <dl><dd><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!+2!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>k</mi> <mo>!</mo> <mo>+</mo> <mn>2</mn> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =k!+2!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f1061ffc1ceeb8b8ddc0160b637a9610480c7aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.961ex; height:2.343ex;" alt="{\displaystyle =k!+2!}"></span></dd> <dd><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(k-1)!+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =k(k-1)!+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc0a1eb3dd6e2bd2344e4efe51792f41579d5691" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.338ex; height:2.843ex;" alt="{\displaystyle =k(k-1)!+2}"></span></dd> <dd><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 =d_{1}b+d_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>b</mi> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =d_{1}b+d_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/967042237b16508b526e288bd1d546d78f6cd508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.818ex; height:2.509ex;" alt="{\displaystyle =d_{1}b+d_{0}}"></span></dd> <dd><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =n_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efbc90a9e47ab2c90ce989782ebaf2d48a42d713" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.902ex; height:2.009ex;" alt="{\displaystyle =n_{2}}"></span></dd></dl></dd></dl> <p>Thus <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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840e456e3058bc0be28e5cf653b170cdbfcc3be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{2}}"></span> is a factorion 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 F_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b151118e130ddb3d3eea064479c152bd16738f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.432ex; height:2.509ex;" alt="{\displaystyle F_{b}}"></span> for 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>. </p> </div> <table class="wikitable"> <caption>Factorions </caption> <tbody><tr> <th><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> </th> <th><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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> </th> <th><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee784b70e772f55ede5e6e0bdc929994bff63413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{1}}"></span> </th> <th><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840e456e3058bc0be28e5cf653b170cdbfcc3be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{2}}"></span> </th></tr> <tr> <td>4</td> <td><a href="/wiki/Base-6" class="mw-redirect" title="Base-6">6</a></td> <td>41</td> <td>42 </td></tr> <tr> <td>5</td> <td>24</td> <td>51</td> <td>52 </td></tr> <tr> <td>6</td> <td>120</td> <td>61</td> <td>62 </td></tr> <tr> <td>7</td> <td>720</td> <td>71</td> <td>72 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="b_=_k!_−_k_+_1"><span id="b_.3D_k.21_.E2.88.92_k_.2B_1"></span><i>b</i> = <i>k</i>! − <i>k</i> + 1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorion&action=edit&section=4" title="Edit section: b = k! − k + 1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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> be a positive integer and the number base <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 b=k!-k+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>k</mi> <mo>!</mo> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=k!-k+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6038bb0baf89a9f77c8057cfa30192eef6d21e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.009ex; height:2.343ex;" alt="{\displaystyle b=k!-k+1}"></span>. Then: </p> <ul><li><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_{1}=b+k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}=b+k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f84472e0a0961822d2b85eb071da927542e04ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.597ex; height:2.509ex;" alt="{\displaystyle n_{1}=b+k}"></span> is a factorion 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 \operatorname {SFD} _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf7993d3f73b0455160ef7b3bdb28fd1c2a8a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.524ex; height:2.509ex;" alt="{\displaystyle \operatorname {SFD} _{b}}"></span> for 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>.</li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p>Let the digits 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 n_{1}=d_{1}b+d_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>b</mi> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}=d_{1}b+d_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79db0a62586cdc09761972dd31fc529272557487" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.912ex; height:2.509ex;" alt="{\displaystyle n_{1}=d_{1}b+d_{0}}"></span> be <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 d_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56b088c47dfeccdfe8268e53843cc965c20294c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.524ex; height:2.509ex;" alt="{\displaystyle d_{1}=1}"></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 d_{0}=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{0}=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d214baa8925472784cce41ee803dae6b0040a192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.573ex; height:2.509ex;" alt="{\displaystyle d_{0}=k}"></span>. Then </p> <dl><dd><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 \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SFD</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>!</mo> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/816fb5ff46337bb0256fc10c537c3f3792a049e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.541ex; height:2.843ex;" alt="{\displaystyle \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}"></span> <dl><dd><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!+k!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>1</mn> <mo>!</mo> <mo>+</mo> <mi>k</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =1!+k!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0cf1e9b3e30d4e19f9e8ece9791ffef17c05d8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.961ex; height:2.343ex;" alt="{\displaystyle =1!+k!}"></span></dd> <dd><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!+1-k+k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>k</mi> <mo>!</mo> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =k!+1-k+k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bc5d0c42031518a7a6b76fd2dbc618d097da872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.417ex; height:2.343ex;" alt="{\displaystyle =k!+1-k+k}"></span></dd> <dd><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(k!-k+1)+k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>1</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>!</mo> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =1(k!-k+1)+k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ecea2553fd45418b9c78ed5609fc94898995c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.389ex; height:2.843ex;" alt="{\displaystyle =1(k!-k+1)+k}"></span></dd> <dd><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 =d_{1}b+d_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>b</mi> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =d_{1}b+d_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/967042237b16508b526e288bd1d546d78f6cd508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.818ex; height:2.509ex;" alt="{\displaystyle =d_{1}b+d_{0}}"></span></dd> <dd><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =n_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09106da232d16dab1ce5d03347e413118cdb15de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.902ex; height:2.009ex;" alt="{\displaystyle =n_{1}}"></span></dd></dl></dd></dl> <p>Thus <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee784b70e772f55ede5e6e0bdc929994bff63413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{1}}"></span> is a factorion 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 F_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b151118e130ddb3d3eea064479c152bd16738f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.432ex; height:2.509ex;" alt="{\displaystyle F_{b}}"></span> for 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>. </p> </div> <table class="wikitable"> <caption>Factorions </caption> <tbody><tr> <th><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> </th> <th><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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> </th> <th><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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee784b70e772f55ede5e6e0bdc929994bff63413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{1}}"></span> </th></tr> <tr> <td>3</td> <td><a href="/wiki/Base-4" class="mw-redirect" title="Base-4">4</a></td> <td>13 </td></tr> <tr> <td>4</td> <td>21</td> <td>14 </td></tr> <tr> <td>5</td> <td>116</td> <td>15 </td></tr> <tr> <td>6</td> <td>715</td> <td>16 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Table_of_factorions_and_cycles_of_SFDb">Table of factorions and cycles of <span class="texhtml"><span class="texhtml">SFD<sub><i>b</i></sub></span></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorion&action=edit&section=5" title="Edit section: Table of factorions and cycles of SFDb"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All numbers are represented in base <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. </p> <table class="wikitable" border="1"> <tbody><tr> <th>Base <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> </th> <th>Nontrivial factorion (<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\neq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≠</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\neq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/096f0036b76638006d76cde5ce49aa80d2a9abf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.656ex; height:2.676ex;" alt="{\displaystyle n\neq 1}"></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 n\neq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≠</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\neq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23ad23a076eeb45aa95aee748953672b6149c165" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.656ex; height:2.676ex;" alt="{\displaystyle n\neq 2}"></span>)<sup id="cite_ref-A193163_10-0" class="reference"><a href="#cite_note-A193163-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </th> <th>Cycles </th></tr> <tr> <td>2</td> <td><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 \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span></td> <td><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 \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span> </td></tr> <tr> <td>3</td> <td><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 \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span></td> <td><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 \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span> </td></tr> <tr> <td>4</td> <td>13</td> <td>3 → 12 → 3 </td></tr> <tr> <td>5</td> <td>144</td> <td><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 \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span> </td></tr> <tr> <td>6</td> <td>41, 42</td> <td><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 \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span> </td></tr> <tr> <td>7</td> <td><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 \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span></td> <td>36 → 2055 → 465 → 2343 → 53 → 240 → 36 </td></tr> <tr> <td>8</td> <td><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 \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span></td> <td> <p>3 → 6 → 1320 → 12 </p><p>175 → 12051 → 175 </p> </td></tr> <tr> <td>9</td> <td>62558</td> <td> </td></tr> <tr> <td>10</td> <td>145, 40585</td> <td> <p>871 → 45361 → 871<sup id="cite_ref-A254499_9-1" class="reference"><a href="#cite_note-A254499-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>872 → 45362 → 872<sup id="cite_ref-A214285_8-1" class="reference"><a href="#cite_note-A214285-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> </td></tr></tbody></table> <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=Factorion&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Arithmetic_dynamics#Other_areas_in_which_number_theory_and_dynamics_interact" title="Arithmetic dynamics">Arithmetic dynamics</a></li> <li><a href="/wiki/Dudeney_number" title="Dudeney number">Dudeney number</a></li> <li><a href="/wiki/Happy_number" title="Happy number">Happy number</a></li> <li><a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's constant</a></li> <li><a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar number</a></li> <li><a href="/wiki/Meertens_number" title="Meertens number">Meertens number</a></li> <li><a href="/wiki/Narcissistic_number" title="Narcissistic number">Narcissistic number</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></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product 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=Factorion&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: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"><ol class="references"> <li id="cite_note-A014080-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-A014080_1-0">^</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" class="citation cs2">Sloane, Neil, <a rel="nofollow" class="external text" href="https://oeis.org/A014080">"A014080"</a>, <i><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=A014080&rft.aulast=Sloane&rft.aufirst=Neil&rft_id=https%3A%2F%2Foeis.org%2FA014080&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorion" class="Z3988"></span></span> </li> <li id="cite_note-Gardner-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gardner_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardner1978" class="citation cs2">Gardner, Martin (1978), "Factorial Oddities", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RcnbvQEACAAJ&q=Mathematical+Magic+Show:+More+Puzzles,+Games,+Diversions,+Illusions+and+Other+Mathematical+Sleight-Of-Mind"><i>Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-Of-Mind</i></a>, Vintage Books, pp. 61 and 64, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780394726236" title="Special:BookSources/9780394726236"><bdi>9780394726236</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Factorial+Oddities&rft.btitle=Mathematical+Magic+Show%3A+More+Puzzles%2C+Games%2C+Diversions%2C+Illusions+and+Other+Mathematical+Sleight-Of-Mind&rft.pages=61+and+64&rft.pub=Vintage+Books&rft.date=1978&rft.isbn=9780394726236&rft.aulast=Gardner&rft.aufirst=Martin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRcnbvQEACAAJ%26q%3DMathematical%2BMagic%2BShow%3A%2BMore%2BPuzzles%2C%2BGames%2C%2BDiversions%2C%2BIllusions%2Band%2BOther%2BMathematical%2BSleight-Of-Mind&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorion" class="Z3988"></span></span> </li> <li id="cite_note-Madachy-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Madachy_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMadachy1979" class="citation cs2">Madachy, Joseph S. (1979), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UvpUAAAAYAAJ&q=Madachy%27s+Mathematical+Recreations"><i>Madachy's Mathematical Recreations</i></a>, Dover Publications, p. 167, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486237626" title="Special:BookSources/9780486237626"><bdi>9780486237626</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Madachy%27s+Mathematical+Recreations&rft.pages=167&rft.pub=Dover+Publications&rft.date=1979&rft.isbn=9780486237626&rft.aulast=Madachy&rft.aufirst=Joseph+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DUvpUAAAAYAAJ%26q%3DMadachy%2527s%2BMathematical%2BRecreations&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorion" class="Z3988"></span></span> </li> <li id="cite_note-Pickover-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pickover_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPickover1995" class="citation cs2">Pickover, Clifford A. (1995), "The Loneliness of the Factorions", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oXIFAAAACAAJ&q=Keys+to+Infinity"><i>Keys to Infinity</i></a>, John Wiley & Sons, pp. 169–171 and 319–320, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471193340" title="Special:BookSources/9780471193340"><bdi>9780471193340</bdi></a> – via Google Books</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Loneliness+of+the+Factorions&rft.btitle=Keys+to+Infinity&rft.pages=169-171+and+319-320&rft.pub=John+Wiley+%26+Sons&rft.date=1995&rft.isbn=9780471193340&rft.aulast=Pickover&rft.aufirst=Clifford+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DoXIFAAAACAAJ%26q%3DKeys%2Bto%2BInfinity&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorion" class="Z3988"></span></span> </li> <li id="cite_note-Gupta-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gupta_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGupta2004" class="citation cs2">Gupta, Shyam S. (2004), "Sum of the Factorials of the Digits of Integers", <i>The Mathematical Gazette</i>, <b>88</b> (512), The Mathematical Association: 258–261, <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/10.1017%2FS0025557200174996">10.1017/S0025557200174996</a></span>, <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/3620841">3620841</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:125854033">125854033</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=Sum+of+the+Factorials+of+the+Digits+of+Integers&rft.volume=88&rft.issue=512&rft.pages=258-261&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125854033%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3620841%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1017%2FS0025557200174996&rft.aulast=Gupta&rft.aufirst=Shyam+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorion" class="Z3988"></span></span> </li> <li id="cite_note-A061602-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-A061602_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane" class="citation cs2">Sloane, Neil, <a rel="nofollow" class="external text" href="https://oeis.org/A061602">"A061602"</a>, <i><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=A061602&rft.aulast=Sloane&rft.aufirst=Neil&rft_id=https%3A%2F%2Foeis.org%2FA061602&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorion" class="Z3988"></span></span> </li> <li id="cite_note-Abbott-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Abbott_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbbott2004" class="citation cs2">Abbott, Steve (2004), "SFD Chains and Factorion Cycles", <i>The Mathematical Gazette</i>, <b>88</b> (512), The Mathematical Association: 261–263, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS002555720017500X">10.1017/S002555720017500X</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/3620842">3620842</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:99976100">99976100</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=SFD+Chains+and+Factorion+Cycles&rft.volume=88&rft.issue=512&rft.pages=261-263&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A99976100%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3620842%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1017%2FS002555720017500X&rft.aulast=Abbott&rft.aufirst=Steve&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorion" class="Z3988"></span></span> </li> <li id="cite_note-A214285-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-A214285_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-A214285_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane" class="citation cs2">Sloane, Neil, <a rel="nofollow" class="external text" href="https://oeis.org/A214285">"A214285"</a>, <i><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=A214285&rft.aulast=Sloane&rft.aufirst=Neil&rft_id=https%3A%2F%2Foeis.org%2FA214285&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorion" class="Z3988"></span></span> </li> <li id="cite_note-A254499-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-A254499_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-A254499_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane" class="citation cs2">Sloane, Neil, <a rel="nofollow" class="external text" href="https://oeis.org/A254499">"A254499"</a>, <i><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=A254499&rft.aulast=Sloane&rft.aufirst=Neil&rft_id=https%3A%2F%2Foeis.org%2FA254499&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorion" class="Z3988"></span></span> </li> <li id="cite_note-A193163-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-A193163_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane" class="citation cs2">Sloane, Neil, <a rel="nofollow" class="external text" href="https://oeis.org/A193163">"A193163"</a>, <i><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=A193163&rft.aulast=Sloane&rft.aufirst=Neil&rft_id=https%3A%2F%2Foeis.org%2FA193163&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorion" class="Z3988"></span></span> </li> </ol></div></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=Factorion&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://mathworld.wolfram.com/Factorion.html">Factorion at Wolfram MathWorld</a></li></ul> <div class="navbox-styles"><style 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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 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_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 class="mw-selflink selflink">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" 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