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class="vector-toc-numb">2</span> <span>Rules for generation</span> </div> </a> <button aria-controls="toc-Rules_for_generation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Rules for generation subsection</span> </button> <ul id="toc-Rules_for_generation-sublist" class="vector-toc-list"> <li id="toc-Thābit_ibn_Qurrah_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thābit_ibn_Qurrah_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Thābit ibn Qurrah theorem</span> </div> </a> <ul id="toc-Thābit_ibn_Qurrah_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Euler's_rule" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Euler's_rule"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Euler's rule</span> </div> </a> <ul id="toc-Euler's_rule-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Regular_pairs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Regular_pairs"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Regular pairs</span> </div> </a> <ul id="toc-Regular_pairs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Twin_amicable_pairs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Twin_amicable_pairs"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Twin amicable pairs</span> </div> </a> <ul id="toc-Twin_amicable_pairs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_results" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Other results</span> </div> </a> <ul id="toc-Other_results-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Generalizations</span> </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Generalizations subsection</span> </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Amicable_tuples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Amicable_tuples"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Amicable tuples</span> </div> </a> <ul id="toc-Amicable_tuples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sociable_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sociable_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Sociable numbers</span> </div> </a> <ul id="toc-Sociable_numbers-sublist" class="vector-toc-list"> <li id="toc-Searching_for_sociable_numbers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Searching_for_sociable_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>Searching for sociable numbers</span> </div> </a> <ul id="toc-Searching_for_sociable_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-References_in_popular_culture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References_in_popular_culture"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References in popular culture</span> </div> </a> <ul id="toc-References_in_popular_culture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Amicable numbers</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 50 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-50" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">50 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A3%D8%B9%D8%AF%D8%A7%D8%AF_%D9%85%D8%AA%D8%AD%D8%A7%D8%A8%D8%A9" title="أعداد متحابة – Arabic" lang="ar" hreflang="ar" data-title="أعداد متحابة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9F%D1%80%D1%8B%D1%85%D1%96%D0%BB%D1%8C%D0%BD%D1%8B%D1%8F_%D0%BB%D1%96%D0%BA%D1%96" title="Прыхільныя лікі – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Прыхільныя лікі" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Niver_karantezus" title="Niver karantezus – Breton" lang="br" hreflang="br" data-title="Niver karantezus" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombres_amics" title="Nombres amics – Catalan" lang="ca" hreflang="ca" data-title="Nombres amics" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Sp%C5%99%C3%A1telen%C3%A1_%C4%8D%C3%ADsla" title="Spřátelená čísla – Czech" lang="cs" hreflang="cs" data-title="Spřátelená čísla" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Venskabstal" title="Venskabstal – Danish" lang="da" hreflang="da" data-title="Venskabstal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Befreundete_Zahlen" title="Befreundete Zahlen – German" lang="de" hreflang="de" data-title="Befreundete Zahlen" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmeros_amigos" title="Números amigos – Spanish" lang="es" hreflang="es" data-title="Números amigos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Amikaj_nombroj" title="Amikaj nombroj – Esperanto" lang="eo" hreflang="eo" data-title="Amikaj nombroj" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_lagun" title="Zenbaki lagun – Basque" lang="eu" hreflang="eu" data-title="Zenbaki lagun" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D9%85%D9%88%D8%A7%D9%81%D9%82" title="اعداد موافق – Persian" lang="fa" hreflang="fa" data-title="اعداد موافق" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombres_amicaux" title="Nombres amicaux – French" lang="fr" hreflang="fr" data-title="Nombres amicaux" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhreacha_caoimhi%C3%BAla" title="Uimhreacha caoimhiúla – Irish" lang="ga" hreflang="ga" data-title="Uimhreacha caoimhiúla" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmeros_amigos" title="Números amigos – Galician" lang="gl" hreflang="gl" data-title="Números amigos" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B9%9C%ED%99%94%EC%88%98" title="친화수 – Korean" lang="ko" hreflang="ko" data-title="친화수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Prijateljski_broj" title="Prijateljski broj – Croatian" lang="hr" hreflang="hr" data-title="Prijateljski broj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Vingjarnlegar_t%C3%B6lur" title="Vingjarnlegar tölur – Icelandic" lang="is" hreflang="is" data-title="Vingjarnlegar tölur" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numeri_amicabili" title="Numeri amicabili – Italian" lang="it" hreflang="it" data-title="Numeri amicabili" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%99%D7%93%D7%99%D7%93%D7%99%D7%9D" title="מספרים ידידים – Hebrew" lang="he" hreflang="he" data-title="מספרים ידידים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%BE%D1%81%D1%82%D0%B0%D1%81_%D1%81%D0%B0%D0%BD%D0%B4%D0%B0%D1%80" title="Достас сандар – Kazakh" lang="kk" hreflang="kk" data-title="Достас сандар" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numeri_amicabiles" title="Numeri amicabiles – Latin" lang="la" hreflang="la" data-title="Numeri amicabiles" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Draugi%C5%A1kasis_skai%C4%8Dius" title="Draugiškasis skaičius – Lithuanian" lang="lt" hreflang="lt" data-title="Draugiškasis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_amix" title="Numer amix – Lombard" lang="lmo" hreflang="lmo" data-title="Numer amix" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bar%C3%A1ts%C3%A1gos_sz%C3%A1mok" title="Barátságos számok – Hungarian" lang="hu" hreflang="hu" data-title="Barátságos számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B5%81%E0%B4%B9%E0%B5%83%E0%B4%A4%E0%B5%8D_%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF%E0%B4%95%E0%B5%BE" title="സുഹൃത് സംഖ്യകൾ – Malayalam" lang="ml" hreflang="ml" data-title="സുഹൃത് സംഖ്യകൾ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bevriende_getallen" title="Bevriende getallen – Dutch" lang="nl" hreflang="nl" data-title="Bevriende getallen" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%8B%E6%84%9B%E6%95%B0" title="友愛数 – Japanese" lang="ja" hreflang="ja" data-title="友愛数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Vennskapstall" title="Vennskapstall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Vennskapstall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Do%CA%BBst_sonlar" title="Doʻst sonlar – Uzbek" lang="uz" hreflang="uz" data-title="Doʻst sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/N%C3%B9mer_amical" title="Nùmer amical – Piedmontese" lang="pms" hreflang="pms" data-title="Nùmer amical" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_zaprzyja%C5%BAnione" title="Liczby zaprzyjaźnione – Polish" lang="pl" hreflang="pl" data-title="Liczby zaprzyjaźnione" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_amigo" title="Número amigo – Portuguese" lang="pt" hreflang="pt" data-title="Número amigo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Numere_prietene" title="Numere prietene – Romanian" lang="ro" hreflang="ro" data-title="Numere prietene" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D1%80%D1%83%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BD%D1%8B%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Дружественные числа – Russian" lang="ru" hreflang="ru" data-title="Дружественные числа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_miq" title="Numrat miq – Albanian" lang="sq" hreflang="sq" data-title="Numrat miq" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%89%E0%B6%AD_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="ඉත සංඛ්‍යා – Sinhala" lang="si" hreflang="si" data-title="ඉත සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Amicable_numbers" title="Amicable numbers – Simple English" lang="en-simple" hreflang="en-simple" data-title="Amicable numbers" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Priate%C4%BEsk%C3%A9_%C4%8D%C3%ADsla" title="Priateľské čísla – Slovak" lang="sk" hreflang="sk" data-title="Priateľské čísla" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Prijateljsko_%C5%A1tevilo" title="Prijateljsko število – Slovenian" lang="sl" hreflang="sl" data-title="Prijateljsko število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D1%98%D0%B0%D1%82%D0%B5%D1%99%D1%81%D0%BA%D0%B8_%D0%B1%D1%80%D0%BE%D1%98" title="Пријатељски број – Serbian" lang="sr" hreflang="sr" data-title="Пријатељски број" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li 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data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Amicable" redirects here. For the definition, see <a href="https://en.wiktionary.org/wiki/amicable" class="extiw" title="wiktionary:amicable">Wiktionary:amicable</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Friendly_numbers" class="mw-redirect" title="Friendly numbers">friendly numbers</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Amicable_numbers_rods_220_and_284.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Amicable_numbers_rods_220_and_284.png/220px-Amicable_numbers_rods_220_and_284.png" decoding="async" width="220" height="282" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Amicable_numbers_rods_220_and_284.png/330px-Amicable_numbers_rods_220_and_284.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e4/Amicable_numbers_rods_220_and_284.png 2x" data-file-width="393" data-file-height="504" /></a><figcaption>Demonstration with <a href="/wiki/Cuisenaire_rods" title="Cuisenaire rods">Cuisenaire rods</a> of the amicability of the pair of numbers (220,284), the first of the series.</figcaption></figure> <p><b>Amicable numbers</b> are two different <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> related in such a way that the <a href="/wiki/Addition" title="Addition">sum</a> of the <a href="/wiki/Proper_divisor" class="mw-redirect" title="Proper divisor">proper divisors</a> of each is equal to the other number. That is, <i>s</i>(<i>a</i>)=<i>b</i> and <i>s</i>(<i>b</i>)=<i>a</i>, where <i>s</i>(<i>n</i>)=σ(<i>n</i>)-<i>n</i> is equal to the sum of positive divisors of <i>n</i> except <i>n</i> itself (see also <a href="/wiki/Divisor_function" title="Divisor function">divisor function</a>). </p><p>The smallest pair of amicable numbers is (<a href="/wiki/220_(number)" title="220 (number)">220</a>, <a href="/wiki/284_(number)" title="284 (number)">284</a>). They are amicable because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220. </p><p>The first ten amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), and (66928, 66992). (sequence <span class="nowrap external"><a href="//oeis.org/A259180" class="extiw" title="oeis:A259180">A259180</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). (Also see <span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A002025" class="extiw" title="oeis:A002025">A002025</a></span> and <span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A002046" class="extiw" title="oeis:A002046">A002046</a></span>) It is unknown if there are infinitely many pairs of amicable numbers. </p><p>A pair of amicable numbers constitutes an <a href="/wiki/Aliquot_sequence" title="Aliquot sequence">aliquot sequence</a> of <a href="/wiki/Periodic_sequence" title="Periodic sequence">period</a> 2. A related concept is that of a <a href="/wiki/Perfect_number" title="Perfect number">perfect number</a>, which is a number that equals the sum of <i>its own</i> proper divisors, in other words a number which forms an aliquot sequence of period 1. Numbers that are members of an aliquot sequence with period greater than 2 are known as <a href="/wiki/Sociable_number" title="Sociable number">sociable numbers</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-mathematics" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-mathematics">Unsolved problem in mathematics</span>:</div> <div class="unsolved-body">Are there infinitely many amicable numbers?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">(more unsolved problems in mathematics)</a></div> </div> <p>Amicable numbers were known to the <a href="/wiki/Pythagoreanism" title="Pythagoreanism">Pythagoreans</a>, who credited them with many mystical properties. A general formula by which some of these numbers could be derived was invented circa 850 by the <a href="/wiki/Iraqi_people" class="mw-redirect" title="Iraqi people">Iraqi</a> mathematician <a href="/wiki/Th%C4%81bit_ibn_Qurra" title="Thābit ibn Qurra">Thābit ibn Qurra</a> (826–901). Other <a href="/wiki/Arab" class="mw-redirect" title="Arab">Arab</a> mathematicians who studied amicable numbers are <a href="/wiki/Maslamah_Ibn_Ahmad_al-Majriti" class="mw-redirect" title="Maslamah Ibn Ahmad al-Majriti">al-Majriti</a> (died 1007), <a href="/wiki/Ibn_Tahir_al-Baghdadi" class="mw-redirect" title="Ibn Tahir al-Baghdadi">al-Baghdadi</a> (980–1037), and <a href="/wiki/Kam%C4%81l_al-D%C4%ABn_al-F%C4%81ris%C4%AB" title="Kamāl al-Dīn al-Fārisī">al-Fārisī</a> (1260–1320). The <a href="/wiki/Iran" title="Iran">Iranian</a> mathematician <a href="/wiki/Muhammad_Baqir_Yazdi" title="Muhammad Baqir Yazdi">Muhammad Baqir Yazdi</a> (16th century) discovered the pair (9363584, 9437056), though this has often been attributed to <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Much of the work of <a href="/wiki/Mathematics_in_medieval_Islam" class="mw-redirect" title="Mathematics in medieval Islam">Eastern mathematicians</a> in this area has been forgotten. </p><p>Thābit ibn Qurra's formula was rediscovered by <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Fermat</a> (1601–1665) and <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a> (1596–1650), to whom it is sometimes ascribed, and extended by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a> (1707–1783). It was extended further by <a href="/wiki/Walter_Borho" title="Walter Borho">Borho</a> in 1972. Fermat and Descartes also rediscovered pairs of amicable numbers known to Arab mathematicians. Euler also discovered dozens of new pairs.<sup id="cite_ref-Sandifer_2-0" class="reference"><a href="#cite_note-Sandifer-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The second smallest pair, (1184, 1210), was discovered in 1867 by 16-year-old B. Nicolò I. Paganini (not to be confused with the composer and violinist), having been overlooked by earlier mathematicians.<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><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable sortable"> <caption>The first ten amicable pairs </caption> <tbody><tr> <th><i>#</i></th> <th><i>m</i></th> <th><i>n</i> </th></tr> <tr> <td>1</td> <td>220</td> <td>284 </td></tr> <tr> <td>2</td> <td>1,184</td> <td>1,210 </td></tr> <tr> <td>3</td> <td>2,620</td> <td>2,924 </td></tr> <tr> <td>4</td> <td>5,020</td> <td>5,564 </td></tr> <tr> <td>5</td> <td>6,232</td> <td>6,368 </td></tr> <tr> <td>6</td> <td>10,744</td> <td>10,856 </td></tr> <tr> <td>7</td> <td>12,285</td> <td>14,595 </td></tr> <tr> <td>8</td> <td>17,296</td> <td>18,416 </td></tr> <tr> <td>9</td> <td>63,020</td> <td>76,084 </td></tr> <tr> <td>10</td> <td>66,928</td> <td>66,992 </td></tr></tbody></table> <p>There are over 1,000,000,000 known amicable pairs.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Rules_for_generation">Rules for generation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=2" title="Edit section: Rules for generation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While these rules do generate some pairs of amicable numbers, many other pairs are known, so these rules are by no means comprehensive. </p><p>In particular, the two rules below produce only even amicable pairs, so they are of no interest for the open problem of finding amicable pairs coprime to 210 = 2·3·5·7, while over 1000 pairs coprime to 30 = 2·3·5 are known [García, Pedersen & te Riele (2003), Sándor & Crstici (2004)]. </p> <div class="mw-heading mw-heading3"><h3 id="Thābit_ibn_Qurrah_theorem"><span id="Th.C4.81bit_ibn_Qurrah_theorem"></span>Thābit ibn Qurrah theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=3" title="Edit section: Thābit ibn Qurrah theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Thābit ibn Qurrah theorem</b> is a method for discovering amicable numbers invented in the 9th century by the <a href="/wiki/Arab" class="mw-redirect" title="Arab">Arab</a> <a href="/wiki/Mathematician" title="Mathematician">mathematician</a> <a href="/wiki/Th%C4%81bit_ibn_Qurrah" class="mw-redirect" title="Thābit ibn Qurrah">Thābit ibn Qurrah</a>.<sup id="cite_ref-Rashed_6-0" class="reference"><a href="#cite_note-Rashed-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>It states that if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p&=3\times 2^{n-1}-1,\\q&=3\times 2^{n}-1,\\r&=9\times 2^{2n-1}-1,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>9</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p&=3\times 2^{n-1}-1,\\q&=3\times 2^{n}-1,\\r&=9\times 2^{2n-1}-1,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22cf5a1c0e17e9216ff623696453fe947fb67b6b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.975ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}p&=3\times 2^{n-1}-1,\\q&=3\times 2^{n}-1,\\r&=9\times 2^{2n-1}-1,\end{aligned}}}"></span> </p><p>where <span class="texhtml"><i>n</i> > 1</span> is an <a href="/wiki/Integer" title="Integer">integer</a> and <span class="texhtml mvar" style="font-style:italic;">p, q, r</span> are <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, then <span class="texhtml">2<sup><i>n</i></sup> × <i>p</i> × <i>q</i></span> and <span class="texhtml">2<sup><i>n</i></sup> × <i>r</i></span> are a pair of amicable numbers. This formula gives the pairs <span class="texhtml">(220, 284)</span> for <span class="texhtml"><i>n</i> = 2</span>, <span class="texhtml">(17296, 18416)</span> for <span class="texhtml"><i>n</i> = 4</span>, and <span class="texhtml">(9363584, 9437056)</span> for <span class="texhtml"><i>n</i> = 7</span>, but no other such pairs are known. Numbers of the form <span class="texhtml">3 × 2<sup><i>n</i></sup> − 1</span> are known as <a href="/wiki/Thabit_number" title="Thabit number">Thabit numbers</a>. In order for Ibn Qurrah's formula to produce an amicable pair, two consecutive Thabit numbers must be prime; this severely restricts the possible values of <span class="texhtml mvar" style="font-style:italic;">n</span>. </p><p>To establish the theorem, Thâbit ibn Qurra proved nine <a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">lemmas</a> divided into two groups. The first three lemmas deal with the determination of the aliquot parts of a <a href="/wiki/Natural_integer" class="mw-redirect" title="Natural integer">natural integer</a>. The second group of lemmas deals more specifically with the formation of perfect, abundant and deficient numbers.<sup id="cite_ref-Rashed_6-1" class="reference"><a href="#cite_note-Rashed-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Euler's_rule"><span id="Euler.27s_rule"></span>Euler's rule</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=4" title="Edit section: Euler's rule"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Euler's rule</i> is a generalization of the Thâbit ibn Qurra theorem. It states that if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p&=(2^{n-m}+1)\times 2^{m}-1,\\q&=(2^{n-m}+1)\times 2^{n}-1,\\r&=(2^{n-m}+1)^{2}\times 2^{m+n}-1,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p&=(2^{n-m}+1)\times 2^{m}-1,\\q&=(2^{n-m}+1)\times 2^{n}-1,\\r&=(2^{n-m}+1)^{2}\times 2^{m+n}-1,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b26f04e045305b364fc47afc000537d75d12f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:29.58ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}p&=(2^{n-m}+1)\times 2^{m}-1,\\q&=(2^{n-m}+1)\times 2^{n}-1,\\r&=(2^{n-m}+1)^{2}\times 2^{m+n}-1,\end{aligned}}}"></span> where <span class="texhtml"><i>n</i> > <i>m</i> > 0</span> are <a href="/wiki/Integer" title="Integer">integers</a> and <span class="texhtml mvar" style="font-style:italic;">p, q, r</span> are <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, then <span class="texhtml">2<sup><i>n</i></sup> × <i>p</i> × <i>q</i></span> and <span class="texhtml">2<sup><i>n</i></sup> × <i>r</i></span> are a pair of amicable numbers. Thābit ibn Qurra's theorem corresponds to the case <span class="texhtml"><i>m</i> = <i>n</i> − 1</span>. Euler's rule creates additional amicable pairs for <span class="texhtml">(<i>m</i>,<i>n</i>) = (1,8), (29,40)</span> with no others being known. Euler (1747 & 1750) overall found 58 new pairs increasing the number of pairs that were then known to 61.<sup id="cite_ref-Sandifer_2-1" class="reference"><a href="#cite_note-Sandifer-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Regular_pairs">Regular pairs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=5" title="Edit section: Regular pairs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let (<span class="texhtml mvar" style="font-style:italic;">m</span>, <span class="texhtml mvar" style="font-style:italic;">n</span>) be a pair of amicable numbers with <span class="texhtml"><i>m</i> < <i>n</i></span>, and write <span class="texhtml"><i>m</i> = <i>gM</i></span> and <span class="texhtml"><i>n</i> = <i>gN</i></span> where <span class="texhtml mvar" style="font-style:italic;">g</span> is the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span>. If <span class="texhtml mvar" style="font-style:italic;">M</span> and <span class="texhtml mvar" style="font-style:italic;">N</span> are both <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> to <span class="texhtml mvar" style="font-style:italic;">g</span> and <a href="/wiki/Square-free_integer" title="Square-free integer">square free</a> then the pair (<span class="texhtml mvar" style="font-style:italic;">m</span>, <span class="texhtml mvar" style="font-style:italic;">n</span>) is said to be <b>regular</b> (sequence <span class="nowrap external"><a href="//oeis.org/A215491" class="extiw" title="oeis:A215491">A215491</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>); otherwise, it is called <b>irregular</b> or <b>exotic</b>. If (<span class="texhtml mvar" style="font-style:italic;">m</span>, <span class="texhtml mvar" style="font-style:italic;">n</span>) is regular and <span class="texhtml mvar" style="font-style:italic;">M</span> and <span class="texhtml mvar" style="font-style:italic;">N</span> have <span class="texhtml mvar" style="font-style:italic;">i</span> and <span class="texhtml mvar" style="font-style:italic;">j</span> prime factors respectively, then <span class="texhtml">(<i>m</i>, <i>n</i>)</span> is said to be of <b>type</b> <span class="texhtml">(<i>i</i>, <i>j</i>)</span>. </p><p>For example, with <span class="texhtml">(<i>m</i>, <i>n</i>) = (220, 284)</span>, the greatest common divisor is <span class="texhtml">4</span> and so <span class="texhtml"><i>M</i> = 55</span> and <span class="texhtml"><i>N</i> = 71</span>. Therefore, <span class="texhtml">(220, 284)</span> is regular of type <span class="texhtml">(2, 1)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Twin_amicable_pairs">Twin amicable pairs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=6" title="Edit section: Twin amicable pairs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An amicable pair <span class="texhtml">(<i>m</i>, <i>n</i>)</span> is twin if there are no integers between <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> belonging to any other amicable pair (sequence <span class="nowrap external"><a href="//oeis.org/A273259" class="extiw" title="oeis:A273259">A273259</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Other_results">Other results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=7" title="Edit section: Other results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In every known case, the numbers of a pair are either both <a href="/wiki/Even_and_odd_numbers" class="mw-redirect" title="Even and odd numbers">even</a> or both odd. It is not known whether an even-odd pair of amicable numbers exists, but if it does, the even number must either be a square number or twice one, and the odd number must be a square number. However, amicable numbers where the two members have different smallest prime factors do exist: there are seven such pairs known.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Also, every known pair shares at least one common prime <a href="/wiki/Divisor" title="Divisor">factor</a>. It is not known whether a pair of <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> amicable numbers exists, though if any does, the <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> of the two must be greater than 10<sup>65</sup>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Also, a pair of co-prime amicable numbers cannot be generated by Thabit's formula (above), nor by any similar formula. </p><p>In 1955 <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> showed that the density of amicable numbers, relative to the positive integers, was 0.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>In 1968 <a href="/wiki/Martin_Gardner" title="Martin Gardner">Martin Gardner</a> noted that most even amicable pairs sumsdivisible by 9,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and that a rule for characterizing the exceptions (sequence <span class="nowrap external"><a href="//oeis.org/A291550" class="extiw" title="oeis:A291550">A291550</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) was obtained.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>According to the sum of amicable pairs conjecture, as the number of the <a href="//oeis.org/A360054" class="extiw" title="oeis:A360054">amicable</a> numbers approaches infinity, the percentage of the sums of the amicable pairs divisible by ten approaches 100% (sequence <span class="nowrap external"><a href="//oeis.org/A291422" class="extiw" title="oeis:A291422">A291422</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). Although all amicable pairs up to 10,000 are even pairs, the proportion of odd amicable pairs increases steadily towards higher numbers, and presumably there are more of them than of the even amicable pairs (<a href="//oeis.org/A360054" class="extiw" title="oeis:A360054">A360054</a> in <a href="//oeis.org/" class="extiw" title="oeis:">OEIS</a>). </p><p><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integer</a> amicable pairs exist,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> e.g. s(8008+3960i) = 4232-8280i and s(4232-8280i) = 8008+3960i.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=8" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Amicable_tuples">Amicable tuples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=9" title="Edit section: Amicable tuples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Amicable 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 (m,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/274d4857135a7d28a94ba9ee8135779615084d43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.278ex; height:2.843ex;" alt="{\displaystyle (m,n)}"></span> satisfy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (m)-m=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (m)-m=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb994a410d7d81f2cc7c84b3f91e780d18f5611d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.553ex; height:2.843ex;" alt="{\displaystyle \sigma (m)-m=n}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (n)-n=m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>n</mi> <mo>=</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (n)-n=m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f941703fd650563f3911655f428e67292149721b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.908ex; height:2.843ex;" alt="{\displaystyle \sigma (n)-n=m}"></span> which can be written together as <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 \sigma (m)=\sigma (n)=m+n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (m)=\sigma (n)=m+n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0444bb86381dacbbd07bdb901eb9495a5575b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.185ex; height:2.843ex;" alt="{\displaystyle \sigma (m)=\sigma (n)=m+n}"></span>. This can be generalized to larger tuples, say <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n_{1},n_{2},\ldots ,n_{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n_{1},n_{2},\ldots ,n_{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103e6e5955afe510f46e4e8b4d146b4b20e22882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.403ex; height:2.843ex;" alt="{\displaystyle (n_{1},n_{2},\ldots ,n_{k})}"></span>, where we require </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 \sigma (n_{1})=\sigma (n_{2})=\dots =\sigma (n_{k})=n_{1}+n_{2}+\dots +n_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (n_{1})=\sigma (n_{2})=\dots =\sigma (n_{k})=n_{1}+n_{2}+\dots +n_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf12c4c040f2abc632e8605b65b0511bb38fc18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.541ex; height:2.843ex;" alt="{\displaystyle \sigma (n_{1})=\sigma (n_{2})=\dots =\sigma (n_{k})=n_{1}+n_{2}+\dots +n_{k}}"></span></dd></dl> <p>For example, (1980, 2016, 2556) is an <a href="/wiki/Amicable_triple" title="Amicable triple">amicable triple</a> (sequence <span class="nowrap external"><a href="//oeis.org/A125490" class="extiw" title="oeis:A125490">A125490</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), and (3270960, 3361680, 3461040, 3834000) is an amicable quadruple (sequence <span class="nowrap external"><a href="//oeis.org/A036471" class="extiw" title="oeis:A036471">A036471</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p><p>Amicable <a href="/wiki/Multiset" title="Multiset">multisets</a> are defined analogously and generalizes this a bit further (sequence <span class="nowrap external"><a href="//oeis.org/A259307" class="extiw" title="oeis:A259307">A259307</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Sociable_numbers">Sociable numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=10" title="Edit section: Sociable numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Sociable_number" title="Sociable number">Sociable number</a></div> <p>Sociable numbers are the numbers in cyclic lists of numbers (with a length greater than 2) where each number is the sum of the proper divisors of the preceding number. For example, <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 1264460\mapsto 1547860\mapsto 1727636\mapsto 1305184\mapsto 1264460\mapsto \dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1264460</mn> <mo stretchy="false">↦</mo> <mn>1547860</mn> <mo stretchy="false">↦</mo> <mn>1727636</mn> <mo stretchy="false">↦</mo> <mn>1305184</mn> <mo stretchy="false">↦</mo> <mn>1264460</mn> <mo stretchy="false">↦</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1264460\mapsto 1547860\mapsto 1727636\mapsto 1305184\mapsto 1264460\mapsto \dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9dac989a1e08d7c84e40c5be9c83100948aae2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:61.48ex; height:2.176ex;" alt="{\displaystyle 1264460\mapsto 1547860\mapsto 1727636\mapsto 1305184\mapsto 1264460\mapsto \dots }"></span> are sociable numbers of order 4. </p> <div class="mw-heading mw-heading4"><h4 id="Searching_for_sociable_numbers">Searching for sociable numbers</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=11" title="Edit section: Searching for sociable numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Aliquot_sequence" title="Aliquot sequence">aliquot sequence</a> can be represented as a <a href="/wiki/Directed_graph" title="Directed graph">directed graph</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 G_{n,s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{n,s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb647f8d64df5cc70abfd55ff8b9029b5415b65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.274ex; height:2.843ex;" alt="{\displaystyle G_{n,s}}"></span>, for a given integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d588236e5ce92331acd2415671cbab71e90cd985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.111ex; height:2.843ex;" alt="{\displaystyle s(k)}"></span> denotes the sum of the proper divisors of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Cycle_(graph_theory)" title="Cycle (graph theory)">Cycles</a> in <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 G_{n,s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{n,s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feb647f8d64df5cc70abfd55ff8b9029b5415b65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.274ex; height:2.843ex;" alt="{\displaystyle G_{n,s}}"></span> represent <a href="/wiki/Sociable_numbers" class="mw-redirect" title="Sociable numbers">sociable numbers</a> within the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [1,n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,n]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c79af450e22e8fd23f28e6be4cb23a47b24c1ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.885ex; height:2.843ex;" alt="{\displaystyle [1,n]}"></span>. Two special cases are loops that represent <a href="/wiki/Perfect_numbers" class="mw-redirect" title="Perfect numbers">perfect numbers</a> and cycles of length two that represent <a href="/wiki/Amicable_pairs" class="mw-redirect" title="Amicable pairs">amicable pairs</a>. </p> <div class="mw-heading mw-heading2"><h2 id="References_in_popular_culture">References in popular culture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=12" title="Edit section: References in popular culture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Amicable numbers are featured in the novel <i><a href="/wiki/The_Housekeeper_and_the_Professor" title="The Housekeeper and the Professor">The Housekeeper and the Professor</a></i> by <a href="/wiki/Y%C5%8Dko_Ogawa" title="Yōko Ogawa">Yōko Ogawa</a>, and in the <a href="/wiki/The_Professor%27s_Beloved_Equation_(film)" title="The Professor's Beloved Equation (film)">Japanese film</a> based on it.</li> <li><a href="/wiki/Paul_Auster" title="Paul Auster">Paul Auster</a>'s collection of short stories entitled <i>True Tales of American Life</i> contains a story ('Mathematical Aphrodisiac' by Alex Galt) in which amicable numbers play an important role.</li> <li>Amicable numbers are featured briefly in the novel <i>The Stranger House</i> by <a href="/wiki/Reginald_Hill" title="Reginald Hill">Reginald Hill</a>.</li> <li>Amicable numbers are mentioned in the French novel <i><a href="/wiki/The_Parrot%27s_Theorem" title="The Parrot's Theorem">The Parrot's Theorem</a></i> by <a href="/wiki/Denis_Guedj" title="Denis Guedj">Denis Guedj</a>.</li> <li>Amicable numbers are mentioned in the JRPG <i><a href="/wiki/Persona_4_Golden" class="mw-redirect" title="Persona 4 Golden">Persona 4 Golden</a></i>.</li> <li>Amicable numbers are featured in the visual novel <i><a href="/wiki/Rewrite_(visual_novel)" class="mw-redirect" title="Rewrite (visual novel)">Rewrite</a></i>.</li> <li>Amicable numbers (220, 284) are referenced in episode 13 of the 2017 Korean drama <a href="/wiki/Andante_(TV_series)" title="Andante (TV series)">Andante</a>.</li> <li>Amicable numbers are featured in the Greek movie <i><a href="/wiki/The_Other_Me_(2016_film)" title="The Other Me (2016 film)">The Other Me (2016 film)</a></i>.</li> <li>Amicable numbers are discussed in the book <i>Are Numbers Real?</i> by <a href="/wiki/Brian_Clegg_(writer)" title="Brian Clegg (writer)">Brian Clegg</a>.</li> <li>Amicable numbers are mentioned in the 2020 novel <i><a href="/wiki/Apeirogon_(novel)" title="Apeirogon (novel)">Apeirogon</a></i> by <a href="/wiki/Colum_McCann" title="Colum McCann">Colum McCann</a>.</li></ul> <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=Amicable_numbers&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed numbers</a> (quasi-amicable numbers)</li> <li><a href="/wiki/Amicable_triple" title="Amicable triple">Amicable triple</a> - Three-number variation of Amicable numbers.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=14" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFCostello2002" class="citation journal cs1">Costello, Patrick (1 May 2002). <a rel="nofollow" class="external text" href="https://www.ams.org/mcom/2003-72-241/S0025-5718-02-01414-X/S0025-5718-02-01414-X.pdf">"New Amicable Pairs Of Type (2; 2) And Type (3; 2)"</a> <span class="cs1-format">(PDF)</span>. <i>Mathematics of Computation</i>. <b>72</b> (241): 489–497. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-02-01414-X">10.1090/S0025-5718-02-01414-X</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080229172358/http://www.ams.org/mcom/2003-72-241/S0025-5718-02-01414-X/S0025-5718-02-01414-X.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2008-02-29<span class="reference-accessdate">. Retrieved <span class="nowrap">19 April</span> 2007</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=New+Amicable+Pairs+Of+Type+%282%3B+2%29+And+Type+%283%3B+2%29&rft.volume=72&rft.issue=241&rft.pages=489-497&rft.date=2002-05-01&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-02-01414-X&rft.aulast=Costello&rft.aufirst=Patrick&rft_id=https%3A%2F%2Fwww.ams.org%2Fmcom%2F2003-72-241%2FS0025-5718-02-01414-X%2FS0025-5718-02-01414-X.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></span> </li> <li id="cite_note-Sandifer-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Sandifer_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Sandifer_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSandifer2007" class="citation book cs1">Sandifer, C. Edward (2007). <i>How Euler Did It</i>. <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. pp. 49–55. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-563-8" title="Special:BookSources/978-0-88385-563-8"><bdi>978-0-88385-563-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+Euler+Did+It&rft.pages=49-55&rft.pub=Mathematical+Association+of+America&rft.date=2007&rft.isbn=978-0-88385-563-8&rft.aulast=Sandifer&rft.aufirst=C.+Edward&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSprugnoli2005" class="citation web cs1 cs1-prop-foreign-lang-source">Sprugnoli, Renzo (27 September 2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120913033238/http://www.dsi.unifi.it/~resp/media.pdf">"Introduzione alla matematica: La matematica della scuola media"</a> <span class="cs1-format">(PDF)</span> (in Italian). Universita degli Studi di Firenze: Dipartimento di Sistemi e Informatica. p. 59. Archived from <a rel="nofollow" class="external text" href="http://www.dsi.unifi.it/~resp/media.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 13 September 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">21 August</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Introduzione+alla+matematica%3A+La+matematica+della+scuola+media&rft.pages=59&rft.pub=Universita+degli+Studi+di+Firenze%3A+Dipartimento+di+Sistemi+e+Informatica&rft.date=2005-09-27&rft.aulast=Sprugnoli&rft.aufirst=Renzo&rft_id=http%3A%2F%2Fwww.dsi.unifi.it%2F~resp%2Fmedia.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin_Gardner2020" class="citation book cs1"><a href="/wiki/Martin_Gardner" title="Martin Gardner">Martin Gardner</a> (2020) [Originally published in 1977]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kE0FEAAAQBAJ&dq=Nicol%C3%B2+I.+Paganini+mathematician&pg=PA168"><i>Mathematical Magic Show</i></a>. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. p. 168. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781470463588" title="Special:BookSources/9781470463588"><bdi>9781470463588</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230912194538/https://books.google.com/books?id=kE0FEAAAQBAJ&dq=Nicol%C3%B2+I.+Paganini+mathematician&pg=PA168">Archived</a> from the original on 2023-09-12<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-03-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Magic+Show&rft.pages=168&rft.pub=American+Mathematical+Society&rft.date=2020&rft.isbn=9781470463588&rft.au=Martin+Gardner&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkE0FEAAAQBAJ%26dq%3DNicol%25C3%25B2%2BI.%2BPaganini%2Bmathematician%26pg%3DPA168&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChernykh" class="citation web cs1">Chernykh, Sergei. <a rel="nofollow" class="external text" href="https://sech.me/ap/">"Amicable pairs list"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-05-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Amicable+pairs+list&rft.aulast=Chernykh&rft.aufirst=Sergei&rft_id=http%3A%2F%2Fsech.me%2Fap%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></span> </li> <li id="cite_note-Rashed-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rashed_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rashed_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRashed1994" class="citation book cs1">Rashed, Roshdi (1994). <i>The development of Arabic mathematics: between arithmetic and algebra</i>. Vol. 156. Dordrecht, Boston, London: Kluwer Academic Publishers. p. 278,279. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-2565-9" title="Special:BookSources/978-0-7923-2565-9"><bdi>978-0-7923-2565-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+development+of+Arabic+mathematics%3A+between+arithmetic+and+algebra.&rft.place=Dordrecht%2C+Boston%2C+London&rft.pages=278%2C279&rft.pub=Kluwer+Academic+Publishers&rft.date=1994&rft.isbn=978-0-7923-2565-9&rft.aulast=Rashed&rft.aufirst=Roshdi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">See <a href="/wiki/William_Dunham_(mathematician)" title="William Dunham (mathematician)">William Dunham</a> in a video: <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=h-DV26x6n_Q&t=37m">An Evening with Leonhard Euler – YouTube</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160516062654/https://www.youtube.com/watch?v=h-DV26x6n_Q&t=37m">Archived</a> 2016-05-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://sech.me/ap/news.html#20160130">"Amicable pairs news"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210718213137/https://sech.me/ap/news.html#20160130">Archived</a> from the original on 2021-07-18<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-01-31</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Amicable+pairs+news&rft_id=http%3A%2F%2Fsech.me%2Fap%2Fnews.html%2320160130&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHagis1969" class="citation journal cs1">Hagis, Peter, Jr. (1969). "On relatively prime odd amicable numbers". <i>Mathematics of Computation</i>. <b>23</b> (107): 539–543. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2004381">10.2307/2004381</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2004381">2004381</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0246816">0246816</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=On+relatively+prime+odd+amicable+numbers&rft.volume=23&rft.issue=107&rft.pages=539-543&rft.date=1969&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D246816%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2004381%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2004381&rft.aulast=Hagis&rft.aufirst=Peter%2C+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHagis1970" class="citation journal cs1">Hagis, Peter, Jr. (1970). 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Dordrecht: Kluwer Academic. pp. 32–36. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-2546-4" title="Special:BookSources/978-1-4020-2546-4"><bdi>978-1-4020-2546-4</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1079.11001">1079.11001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+number+theory+II&rft.place=Dordrecht&rft.pages=32-36&rft.pub=Kluwer+Academic&rft.date=2004&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1079.11001%23id-name%3DZbl&rft.isbn=978-1-4020-2546-4&rft.aulast=S%C3%A1ndor&rft.aufirst=Jozsef&rft.au=Crstici%2C+Borislav&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWells1987" class="citation book cs1">Wells, D. 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Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AmicablePair.html">"Amicable Pair"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Amicable+Pair&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FAmicablePair.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Thâbit_ibn_Kurrah_Rule"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ThabitibnKurrahRule.html">"Thâbit ibn Kurrah Rule"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Th%C3%A2bit+ibn+Kurrah+Rule&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FThabitibnKurrahRule.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Euler's_Rule"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/EulersRule.html">"Euler's Rule"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Euler%27s+Rule&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FEulersRule.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Amicable_numbers&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFM._GarcíaJ.M._PedersenH.J.J._te_Riele2003" class="citation journal cs1">M. García; J.M. Pedersen; H.J.J. te Riele (2003-07-31). <a rel="nofollow" class="external text" href="http://oai.cwi.nl/oai/asset/4143/04143D.pdf">"Amicable pairs, a survey"</a> <span class="cs1-format">(PDF)</span>. <i>Report MAS-R0307</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Report+MAS-R0307&rft.atitle=Amicable+pairs%2C+a+survey&rft.date=2003-07-31&rft.au=M.+Garc%C3%ADa&rft.au=J.M.+Pedersen&rft.au=H.J.J.+te+Riele&rft_id=http%3A%2F%2Foai.cwi.nl%2Foai%2Fasset%2F4143%2F04143D.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrime" class="citation web cs1">Grime, James. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170716184637/http://www.numberphile.com/videos/220_284.html">"220 and 284 (Amicable Numbers)"</a>. <i>Numberphile</i>. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>. Archived from <a rel="nofollow" class="external text" href="http://www.numberphile.com/videos/220_284.html">the original</a> on 2017-07-16<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-04-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Numberphile&rft.atitle=220+and+284+%28Amicable+Numbers%29&rft.aulast=Grime&rft.aufirst=James&rft_id=http%3A%2F%2Fwww.numberphile.com%2Fvideos%2F220_284.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrime" class="citation web cs1">Grime, James. <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=R2eQVqdUQLI">"MegaFavNumbers - The Even Amicable Numbers Conjecture"</a>. <i>YouTube</i>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211123/R2eQVqdUQLI">Archived</a> from the original on 2021-11-23<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-06-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=YouTube&rft.atitle=MegaFavNumbers+-+The+Even+Amicable+Numbers+Conjecture&rft.aulast=Grime&rft.aufirst=James&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DR2eQVqdUQLI&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoutsoukou-Argyraki2020" class="citation web cs1">Koutsoukou-Argyraki, Angeliki (4 August 2020). <a rel="nofollow" class="external text" href="https://www.isa-afp.org/entries/Amicable_Numbers.html">"Amicable Numbers (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Amicable+Numbers+%28Formal+proof+development+in+Isabelle%2FHOL%2C+Archive+of+Formal+Proofs%29&rft.date=2020-08-04&rft.aulast=Koutsoukou-Argyraki&rft.aufirst=Angeliki&rft_id=https%3A%2F%2Fwww.isa-afp.org%2Fentries%2FAmicable_Numbers.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChernykh" class="citation web cs1">Chernykh, Sergei. <a rel="nofollow" class="external text" href="https://sech.me/ap/">"Amicable pairs list"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-09-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Amicable+pairs+list&rft.aulast=Chernykh&rft.aufirst=Sergei&rft_id=https%3A%2F%2Fsech.me%2Fap%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAmicable+numbers" class="Z3988"></span> (database of all known amicable numbers)</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol 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title="Fundamental theorem of arithmetic">Fundamental theorem of arithmetic</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Lattice_of_the_divisibility_of_60.svg" class="mw-file-description" title="Divisibility of 60"><img alt="Divisibility of 60" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/175px-Lattice_of_the_divisibility_of_60.svg.png" decoding="async" width="175" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/263px-Lattice_of_the_divisibility_of_60.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/350px-Lattice_of_the_divisibility_of_60.svg.png 2x" data-file-width="313" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Factorization forms</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prime_number" title="Prime number">Prime</a></li> <li><a href="/wiki/Composite_number" title="Composite number">Composite</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/Square-free_integer" title="Square-free integer">Square-free</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a 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/Unusual_number" title="Unusual number">Unusual</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constrained divisor sums</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li> <li><a href="/wiki/Unitary_perfect_number" title="Unitary perfect number">Unitary perfect</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With many divisors</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Weird_number" title="Weird number">Weird</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequence</a>-related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li> <li><a class="mw-selflink selflink">Amicable</a> (<a href="/wiki/Amicable_triple" title="Amicable triple">Triple</a>)</li> <li><a href="/wiki/Sociable_number" title="Sociable number">Sociable</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Radix" title="Radix">Base</a>-dependent</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other sets</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Friendly_number#Solitary_numbers" title="Friendly number">Solitary</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Classes_of_natural_numbers" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classes_of_natural_numbers" title="Template:Classes of natural numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classes_of_natural_numbers" title="Template talk:Classes of natural numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classes_of_natural_numbers" title="Special:EditPage/Template:Classes of natural numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Classes_of_natural_numbers" style="font-size:114%;margin:0 4em">Classes of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Exponentiation" title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of 2</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Power of 3</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Power of 10</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cube</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Fourth power</a></li> <li><a href="/wiki/Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">Sixth power</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">Seventh power</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">Eighth power</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Prime_power" title="Prime power">Prime power</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cullen_number" title="Cullen number">Cullen</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li> <li><a href="/wiki/Fermat_number" title="Fermat number">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell</a></li> <li><a href="/wiki/Perrin_number" title="Perrin number">Perrin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_number" title="Amenable number">Amenable</a></li> <li><a href="/wiki/Congruent_number" title="Congruent number">Congruent</a></li> <li><a href="/wiki/Kn%C3%B6del_number" title="Knödel number">Knödel</a></li> <li><a href="/wiki/Riesel_number" title="Riesel number">Riesel</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li> <li><a href="/wiki/Polite_number" title="Polite number">Polite</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li> <li><a href="/wiki/Ulam_number" title="Ulam number">Ulam</a></li> <li><a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li> <li><a href="/wiki/Centered_square_number" title="Centered square number">Centered square</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li> <li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li> <li><a href="/wiki/Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li> <li><a href="/wiki/Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li> <li><a href="/wiki/Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li> <li><a href="/wiki/Star_number" title="Star number">Star</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triangular_number" title="Triangular number">Triangular</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Square_triangular_number" title="Square triangular number">Square triangular</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentatope_number" title="Pentatope number">Pentatope</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_number" title="Bell number">Bell</a></li> <li><a href="/wiki/Cake_number" title="Cake number">Cake</a></li> <li><a href="/wiki/Catalan_number" title="Catalan number">Catalan</a></li> <li><a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind</a></li> <li><a href="/wiki/Delannoy_number" title="Delannoy number">Delannoy</a></li> <li><a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler</a></li> <li><a href="/wiki/Eulerian_number" title="Eulerian number">Eulerian</a></li> <li><a href="/wiki/Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li> <li><a href="/wiki/Lah_number" title="Lah number">Lah</a></li> <li><a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li> <li><a href="/wiki/Lobb_number" title="Lobb number">Lobb</a></li> <li><a href="/wiki/Motzkin_number" title="Motzkin number">Motzkin</a></li> <li><a href="/wiki/Narayana_number" title="Narayana number">Narayana</a></li> <li><a href="/wiki/Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li> <li><a href="/wiki/Schr%C3%B6der_number" title="Schröder number">Schröder</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li> <li><a href="/wiki/Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li> <li><a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li> <li><a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li> <li><a href="/wiki/Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li> <li><a href="/wiki/Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes" style="font-size:114%;margin:0 4em"><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li> <li><a href="/wiki/Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics" style="font-size:114%;margin:0 4em"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Highly_totient_number" title="Highly totient number">Highly totient</a></li> <li><a href="/wiki/Noncototient" title="Noncototient">Noncototient</a></li> <li><a href="/wiki/Nontotient" title="Nontotient">Nontotient</a></li> <li><a href="/wiki/Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li> <li><a href="/wiki/Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">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" 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