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data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_hexagonal_centr%C3%A9" title="Nombre hexagonal centré – French" lang="fr" hreflang="fr" data-title="Nombre hexagonal centré" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_esagonale_centrato" title="Numero esagonale centrato – Italian" lang="it" hreflang="it" data-title="Numero esagonale centrato" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/K%C3%B6z%C3%A9ppontos_hatsz%C3%B6gsz%C3%A1mok" 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Number that represents a hexagon with a dot in the center</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Catan_Universe_fixed_setup.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Catan_Universe_fixed_setup.svg/220px-Catan_Universe_fixed_setup.svg.png" decoding="async" width="220" height="193" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Catan_Universe_fixed_setup.svg/330px-Catan_Universe_fixed_setup.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Catan_Universe_fixed_setup.svg/440px-Catan_Universe_fixed_setup.svg.png 2x" data-file-width="512" data-file-height="448" /></a><figcaption>Centered hexagonal numbers appearing in the <a href="/wiki/Catan" title="Catan">Catan</a> board game:<br />19 land tiles,<br />37 total tiles</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, a <b>centered hexagonal number</b>, or <b>hex number</b>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Deza_2-0" class="reference"><a href="#cite_note-Deza-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is a <a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a> <a href="/wiki/Figurate_number" title="Figurate number">figurate number</a> that represents a <a href="/wiki/Hexagon" title="Hexagon">hexagon</a> with a dot in the center and all other dots surrounding the center dot in a <a href="/wiki/Hexagonal_lattice" title="Hexagonal lattice">hexagonal lattice</a>. The following figures illustrate this arrangement for the first four centered hexagonal numbers: </p> <dl><dd><table style="min-width: 325px;"> <tbody><tr> <th>1</th> <th></th> <th>7</th> <th></th> <th>19</th> <th></th> <th>37 </th></tr> <tr style="text-align:center; color:red; vertical-align:middle;"> <td>+1</td> <td></td> <td>+6</td> <td></td> <td>+12</td> <td></td> <td>+18 </td></tr> <tr style="vertical-align:middle; text-align:center; line-height:1.1em;"> <td><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> </td> <td> </td> <td><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" 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typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> <span typeof="mw:File"><a href="/wiki/File:RedDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/16px-RedDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/24px-RedDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/RedDotX.svg/32px-RedDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> </td></tr></tbody></table></dd></dl> <p>Centered hexagonal numbers should not be confused with <a href="/wiki/Hexagonal_number" title="Hexagonal number">cornered hexagonal numbers</a>, which are figurate numbers in which the associated hexagons share a vertex. </p><p>The sequence of hexagonal numbers starts out as follows (sequence <span class="nowrap external"><a href="//oeis.org/A003215" class="extiw" title="oeis:A003215">A003215</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>): </p> <dl><dd><a href="/wiki/1" title="1">1</a>, <a href="/wiki/7" title="7">7</a>, <a href="/wiki/19_(number)" title="19 (number)">19</a>, <a href="/wiki/37_(number)" title="37 (number)">37</a>, <a href="/wiki/61_(number)" title="61 (number)">61</a>, <a href="/wiki/91_(number)" title="91 (number)">91</a>, <a href="/wiki/127_(number)" title="127 (number)">127</a>, <a href="/wiki/169_(number)" title="169 (number)">169</a>, <a href="/wiki/217_(number)" title="217 (number)">217</a>, <a href="/wiki/271_(number)" title="271 (number)">271</a>, <a href="/wiki/331_(number)" class="mw-redirect" title="331 (number)">331</a>, <a href="/wiki/397_(number)" class="mw-redirect" title="397 (number)">397</a>, 469, 547, 631, 721, 817, 919.</dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formula">Formula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centered_hexagonal_number&action=edit&section=1" title="Edit section: Formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Centered_hexagonal_%3D_1_%2B_6triangular.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Centered_hexagonal_%3D_1_%2B_6triangular.svg/220px-Centered_hexagonal_%3D_1_%2B_6triangular.svg.png" decoding="async" width="220" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Centered_hexagonal_%3D_1_%2B_6triangular.svg/330px-Centered_hexagonal_%3D_1_%2B_6triangular.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Centered_hexagonal_%3D_1_%2B_6triangular.svg/440px-Centered_hexagonal_%3D_1_%2B_6triangular.svg.png 2x" data-file-width="576" data-file-height="496" /></a><figcaption>Dissection of hexagonal number into six triangles with a remainder of one. The triangles can be re-assembled pairwise to give three <a href="/wiki/Parallelogram" title="Parallelogram">parallelograms</a> of <span class="texhtml"><i>n</i>(<i>n</i>−1)</span> dots each.</figcaption></figure> <p>The <span class="texhtml mvar" style="font-style:italic;">n</span>th centered hexagonal number is given by the formula<sup id="cite_ref-Deza_2-1" class="reference"><a href="#cite_note-Deza-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </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 H(n)=n^{3}-(n-1)^{3}=3n(n-1)+1=3n^{2}-3n+1.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>3</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>3</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1.</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(n)=n^{3}-(n-1)^{3}=3n(n-1)+1=3n^{2}-3n+1.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57ab960df54ff06afe4d7eb24757758f416151e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.926ex; height:3.176ex;" alt="{\displaystyle H(n)=n^{3}-(n-1)^{3}=3n(n-1)+1=3n^{2}-3n+1.\,}"></span></dd></dl> <p>Expressing the formula as </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 H(n)=1+6\left({\frac {n(n-1)}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>6</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(n)=1+6\left({\frac {n(n-1)}{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7906504b2c0e8d574dd70018738c85e738ed3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.777ex; height:6.343ex;" alt="{\displaystyle H(n)=1+6\left({\frac {n(n-1)}{2}}\right)}"></span></dd></dl> <p>shows that the centered hexagonal number for <span class="texhtml mvar" style="font-style:italic;">n</span> is 1 more than 6 times the <span class="texhtml">(<i>n</i> − 1)</span>th <a href="/wiki/Triangular_number" title="Triangular number">triangular number</a>. </p><p>In the opposite direction, the <i>index</i> <span class="texhtml mvar" style="font-style:italic;">n</span> corresponding to the centered hexagonal 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 H=H(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=H(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b55116df2a28b1028b2ca5dbe88ad4c0beabb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.43ex; height:2.843ex;" alt="{\displaystyle H=H(n)}"></span> can be calculated using the formula </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 n={\frac {3+{\sqrt {12H-3}}}{6}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>12</mn> <mi>H</mi> <mo>−</mo> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\frac {3+{\sqrt {12H-3}}}{6}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03d0b93fe17b0d54371bc6a4043add7d8b334946" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.306ex; height:5.843ex;" alt="{\displaystyle n={\frac {3+{\sqrt {12H-3}}}{6}}.}"></span></dd></dl> <p>This can be used as a test for whether a number <span class="texhtml mvar" style="font-style:italic;">H</span> is centered hexagonal: it will be if and only if the above expression is an integer. </p> <div class="mw-heading mw-heading2"><h2 id="Recurrence_and_generating_function">Recurrence and generating function</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centered_hexagonal_number&action=edit&section=2" title="Edit section: Recurrence and generating function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The centered hexagonal 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 H(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be4e983a8ea1e65323a9af036edd5d57c51e8c18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.268ex; height:2.843ex;" alt="{\displaystyle H(n)}"></span> satisfy the <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a><sup id="cite_ref-Deza_2-2" class="reference"><a href="#cite_note-Deza-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </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 H(n+1)=H(n)+6n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>6</mn> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(n+1)=H(n)+6n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b06bf50fe6576387ae1395bf2d17653040d7be2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.681ex; height:2.843ex;" alt="{\displaystyle H(n+1)=H(n)+6n.}"></span></dd></dl> <p>From this we can calculate the <a href="/wiki/Generating_function" title="Generating function">generating function</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 F(x)=\sum _{n\geq 0}H(n)x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mi>H</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=\sum _{n\geq 0}H(n)x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f235bfdad7b8759cc958a589b0b492a3a89a983f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.536ex; height:5.676ex;" alt="{\displaystyle F(x)=\sum _{n\geq 0}H(n)x^{n}}"></span>. The generating function satisfies </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 F(x)=x+xF(x)+\sum _{n\geq 2}6nx^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </munder> <mn>6</mn> <mi>n</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=x+xF(x)+\sum _{n\geq 2}6nx^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c6256506f544c90c93c7fd7f5789edcff6c6162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.692ex; height:5.676ex;" alt="{\displaystyle F(x)=x+xF(x)+\sum _{n\geq 2}6nx^{n}.}"></span></dd></dl> <p>The latter term is the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</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 {\frac {6x}{(1-x)^{2}}}-6x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <mi>x</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>6</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {6x}{(1-x)^{2}}}-6x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fdc2a179c10af2150750f84c767bd785bb39eb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.365ex; height:6.009ex;" alt="{\displaystyle {\frac {6x}{(1-x)^{2}}}-6x}"></span>, so we get </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 (1-x)F(x)=x+{\frac {6x}{(1-x)^{2}}}-6x={\frac {x+4x^{2}+x^{3}}{(1-x)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <mi>x</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-x)F(x)=x+{\frac {6x}{(1-x)^{2}}}-6x={\frac {x+4x^{2}+x^{3}}{(1-x)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c80bc0443f1c4f8b1f2b447a3d0628297494f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.53ex; height:6.509ex;" alt="{\displaystyle (1-x)F(x)=x+{\frac {6x}{(1-x)^{2}}}-6x={\frac {x+4x^{2}+x^{3}}{(1-x)^{2}}}}"></span></dd></dl> <p>and end up at </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 F(x)={\frac {x+4x^{2}+x^{3}}{(1-x)^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)={\frac {x+4x^{2}+x^{3}}{(1-x)^{3}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6c44cf0439bd4d9bdef96516513c467dfba09a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.402ex; height:6.509ex;" alt="{\displaystyle F(x)={\frac {x+4x^{2}+x^{3}}{(1-x)^{3}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centered_hexagonal_number&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Visual_proof_centered_hexagonal_numbers_sum.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Visual_proof_centered_hexagonal_numbers_sum.svg/220px-Visual_proof_centered_hexagonal_numbers_sum.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Visual_proof_centered_hexagonal_numbers_sum.svg/330px-Visual_proof_centered_hexagonal_numbers_sum.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Visual_proof_centered_hexagonal_numbers_sum.svg/440px-Visual_proof_centered_hexagonal_numbers_sum.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption><a href="/wiki/Proof_without_words" title="Proof without words">Proof without words</a> of the sum of the first <i>n</i> hex numbers by arranging <i>n</i><sup>3</sup> semitransparent balls in a cube and viewing along a <a href="/wiki/Space_diagonal" title="Space diagonal">space diagonal</a> – colour denotes cube layer and line style denotes hex number</figcaption></figure> <p>In <a href="/wiki/Base_10" class="mw-redirect" title="Base 10">base 10</a> one can notice that the hexagonal numbers' rightmost (least significant) digits follow the pattern 1–7–9–7–1 (repeating with <a href="/wiki/Periodic_sequence" title="Periodic sequence">period</a> 5). This follows from the <a href="/wiki/Triangular_number#Other_properties" title="Triangular number">last digit of the triangle numbers</a> (sequence <span class="nowrap external"><a href="//oeis.org/A008954" class="extiw" title="oeis:A008954">A008954</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) which repeat 0-1-3-1-0 when taken modulo 5. In <a href="/wiki/Base_6" class="mw-redirect" title="Base 6">base 6</a> the rightmost digit is always 1: 1<sub>6</sub>, 11<sub>6</sub>, 31<sub>6</sub>, 101<sub>6</sub>, 141<sub>6</sub>, 231<sub>6</sub>, 331<sub>6</sub>, 441<sub>6</sub>... This follows from the fact that every centered hexagonal number modulo 6 (=10<sub>6</sub>) equals 1. </p><p>The sum of the first <span class="texhtml mvar" style="font-style:italic;">n</span> centered hexagonal numbers is <span class="texhtml"><a href="/wiki/Cube_(algebra)" title="Cube (algebra)"><i>n</i><sup>3</sup></a></span>. That is, centered hexagonal <a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal numbers</a> and <a href="/wiki/Cubic_number" class="mw-redirect" title="Cubic number">cubes</a> are the same numbers, but they represent different shapes. Viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the <a href="/wiki/Figurate_number#Gnomon" title="Figurate number">gnomon</a> of the cubes. (This can be seen geometrically from the diagram.) In particular, <a href="/wiki/Prime_number" title="Prime number">prime</a> centered hexagonal numbers are <a href="/wiki/Cuban_prime" title="Cuban prime">cuban primes</a>. </p><p>The difference between <span class="texhtml">(2<i>n</i>)<sup>2</sup></span> and the <span class="texhtml mvar" style="font-style:italic;">n</span>th centered hexagonal number is a number of the form <span class="texhtml">3<i>n</i><sup>2</sup> + 3<i>n</i> − 1</span>, while the difference between <span class="texhtml">(2<i>n</i> − 1)<sup>2</sup></span> and the <span class="texhtml mvar" style="font-style:italic;">n</span>th centered hexagonal number is a <a href="/wiki/Pronic_number" title="Pronic number">pronic number</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centered_hexagonal_number&action=edit&section=4" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Comparison_optical_telescope_primary_mirrors.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Comparison_optical_telescope_primary_mirrors.svg/220px-Comparison_optical_telescope_primary_mirrors.svg.png" decoding="async" width="220" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Comparison_optical_telescope_primary_mirrors.svg/330px-Comparison_optical_telescope_primary_mirrors.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Comparison_optical_telescope_primary_mirrors.svg/440px-Comparison_optical_telescope_primary_mirrors.svg.png 2x" data-file-width="512" data-file-height="536" /></a><figcaption>Ignoring central holes, the number of mirror segments in several <a href="/wiki/Segmented_mirror" title="Segmented mirror">segmented mirror</a> <a href="/wiki/Telescope" title="Telescope">telescopes</a> are centered hexagonal numbers</figcaption></figure> <p>Many <a href="/wiki/Segmented_mirror" title="Segmented mirror">segmented mirror</a> <a href="/wiki/Reflecting_telescope" title="Reflecting telescope">reflecting telescopes</a> have primary mirrors comprising a centered hexagonal number of segments (neglecting the central segment removed to allow passage of light) to simplify the control system.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Some examples: </p> <table class="wikitable" style="text-align:center;"> <tbody><tr> <th>Telescope</th> <th>Number of<br />segments</th> <th>Number<br />missing</th> <th>Total</th> <th><i>n</i>-th centered<br />hexagonal number </th></tr> <tr> <td align="left"><a href="/wiki/Giant_Magellan_Telescope" title="Giant Magellan Telescope">Giant Magellan Telescope</a></td> <td>7</td> <td>0</td> <td><b>7</b></td> <td>2 </td></tr> <tr> <td align="left"><a href="/wiki/James_Webb_Space_Telescope" title="James Webb Space Telescope">James Webb Space Telescope</a></td> <td>18</td> <td>1</td> <td><b>19</b></td> <td>3 </td></tr> <tr> <td align="left"><a href="/wiki/Gran_Telescopio_Canarias" title="Gran Telescopio Canarias">Gran Telescopio Canarias</a></td> <td>36</td> <td>1</td> <td><b>37</b></td> <td>4 </td></tr> <tr> <td align="left"><a href="/wiki/Guido_Horn_d%27Arturo" title="Guido Horn d'Arturo">Guido Horn d'Arturo</a>'s prototype</td> <td>61</td> <td>0</td> <td><b>61</b></td> <td>5 </td></tr> <tr> <td align="left"><a href="/wiki/Southern_African_Large_Telescope" title="Southern African Large Telescope">Southern African Large Telescope</a></td> <td>91</td> <td>0</td> <td><b>91</b></td> <td>6 </td></tr></tbody></table> <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=Centered_hexagonal_number&action=edit&section=5" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHindin1983" class="citation journal cs1">Hindin, H. J. (1983). "Stars, hexes, triangular numbers and Pythagorean triples". <i>J. Rec. Math</i>. <b>16</b>: 191–193.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Rec.+Math.&rft.atitle=Stars%2C+hexes%2C+triangular+numbers+and+Pythagorean+triples&rft.volume=16&rft.pages=191-193&rft.date=1983&rft.aulast=Hindin&rft.aufirst=H.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentered+hexagonal+number" class="Z3988"></span></span> </li> <li id="cite_note-Deza-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Deza_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Deza_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Deza_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDezaDeza2012" class="citation book cs1"><a href="/wiki/Elena_Deza" title="Elena Deza">Deza, Elena</a>; Deza, M. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cDxYdstLPz4C"><i>Figurate Numbers</i></a>. World Scientific. pp. 47–55. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4355-48-3" title="Special:BookSources/978-981-4355-48-3"><bdi>978-981-4355-48-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Figurate+Numbers&rft.pages=47-55&rft.pub=World+Scientific&rft.date=2012&rft.isbn=978-981-4355-48-3&rft.aulast=Deza&rft.aufirst=Elena&rft.au=Deza%2C+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcDxYdstLPz4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentered+hexagonal+number" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Mast, T. S. and Nelson, J. E. <a rel="nofollow" class="external text" href="https://osti.gov/servlets/purl/6194407"><i>Figure control for a segmented telescope mirror</i></a>. United States: N. p., 1979. Web. doi:10.2172/6194407.</span> </li> </ol></div></div> <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=Centered_hexagonal_number&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/Hexagonal_number" title="Hexagonal number">Hexagonal number</a></li> <li><a href="/wiki/Magic_hexagon" title="Magic hexagon">Magic hexagon</a></li> <li><a href="/wiki/Star_number" title="Star number">Star number</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline 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nonagonal numbers</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal numbers</a></li> <li><a href="/wiki/Star_number" title="Star number">Star numbers</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 numbers</a></li> <li><a href="/wiki/Square_number" title="Square number">Square numbers</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal numbers</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal numbers</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal numbers</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal 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<ul><li><a href="/wiki/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral numbers</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube numbers</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral numbers</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral numbers</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cube numbers</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral numbers</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral numbers</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral numbers</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedral_number" title="Tetrahedral number">Tetrahedral numbers</a></li> <li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal numbers</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentatope_number" title="Pentatope number">Pentatope numbers</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular numbers</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic numbers</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Higher <a href="/wiki/Dimension" title="Dimension">dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="non-centered" scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fifth_power_(algebra)" title="Fifth power (algebra)">5-hypercube numbers</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">6-hypercube numbers</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">7-hypercube numbers</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">8-hypercube numbers</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Classes_of_natural_numbers" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classes_of_natural_numbers" title="Template:Classes of natural numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classes_of_natural_numbers" title="Template talk:Classes of natural numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classes_of_natural_numbers" title="Special:EditPage/Template:Classes of natural numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Classes_of_natural_numbers" style="font-size:114%;margin:0 4em">Classes of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Exponentiation" title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of 2</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Power of 3</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Power of 10</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cube</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Fourth power</a></li> <li><a href="/wiki/Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">Sixth power</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">Seventh power</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">Eighth power</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Prime_power" title="Prime power">Prime power</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cullen_number" title="Cullen number">Cullen</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li> <li><a href="/wiki/Fermat_number" title="Fermat number">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell</a></li> <li><a href="/wiki/Perrin_number" title="Perrin number">Perrin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_number" title="Amenable number">Amenable</a></li> <li><a href="/wiki/Congruent_number" title="Congruent number">Congruent</a></li> <li><a href="/wiki/Kn%C3%B6del_number" title="Knödel number">Knödel</a></li> <li><a href="/wiki/Riesel_number" title="Riesel number">Riesel</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li> <li><a href="/wiki/Polite_number" title="Polite number">Polite</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li> <li><a href="/wiki/Ulam_number" title="Ulam number">Ulam</a></li> <li><a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li> <li><a href="/wiki/Centered_square_number" title="Centered square number">Centered square</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li> <li><a class="mw-selflink selflink">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 href="/wiki/Factorion" title="Factorion">Factorion</a></li> <li><a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar</a></li> <li><a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's constant</a></li> <li><a href="/wiki/Keith_number" title="Keith number">Keith</a></li> <li><a href="/wiki/Lychrel_number" title="Lychrel number">Lychrel</a></li> <li><a href="/wiki/Narcissistic_number" title="Narcissistic number">Narcissistic</a></li> <li><a href="/wiki/Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li> <li><a href="/wiki/Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a> <ul><li><a href="/wiki/Happy_number" title="Happy number">Happy</a></li></ul></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Automorphic_number" title="Automorphic number">Automorphic</a> <ul><li><a href="/wiki/Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_number" title="Palindromic number">Palindromic</a></li> <li><a href="/wiki/Pandigital_number" title="Pandigital number">Pandigital</a></li> <li><a href="/wiki/Repdigit" title="Repdigit">Repdigit</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit</a></li> <li><a href="/wiki/Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li> <li><a href="/wiki/Undulating_number" title="Undulating number">Undulating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="/wiki/Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cyclic_number" title="Cyclic number">Cyclic</a></li> <li><a href="/wiki/Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li> <li><a href="/wiki/Parasitic_number" title="Parasitic number">Parasitic</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Transposable_integer" title="Transposable integer">Transposable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li> <li><a href="/wiki/Vampire_number" title="Vampire number">Vampire</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Friedman_number" title="Friedman number">Friedman</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evil_number" title="Evil number">Evil</a></li> <li><a href="/wiki/Odious_number" title="Odious number">Odious</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve" style="font-size:114%;margin:0 4em">Generated via a <a href="/wiki/Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Generation_of_primes" title="Generation of primes">Prime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pancake_sorting" title="Pancake sorting">Pancake number</a></li> <li><a href="/wiki/Sorting_number" title="Sorting number">Sorting number</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aronson%27s_sequence" title="Aronson's sequence">Aronson's sequence</a></li> <li><a href="/wiki/Ban_number" title="Ban number">Ban</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2" style="font-weight:bold;"><div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span> <a href="/wiki/Portal:Mathematics" 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