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class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-最小的k重完全數" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#最小的k重完全數"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>最小的<i>k</i>重完全數</span> </div> </a> <ul id="toc-最小的k重完全數-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考資料" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參考資料"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>參考資料</span> </div> </a> <ul id="toc-參考資料-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部連結" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部連結"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>外部連結</span> </div> </a> <ul id="toc-外部連結-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="目录" 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="开关目录" > <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">开关目录</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">多重完全數</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="前往另一种语言写成的文章。13种语言可用" > <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-13" 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">13种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/N%C3%A1sobn%C4%9B_dokonal%C3%A9_%C4%8D%C3%ADslo" title="Násobně dokonalé číslo – 捷克语" lang="cs" hreflang="cs" data-title="Násobně dokonalé číslo" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Multiply_perfect_number" title="Multiply perfect number – 英语" lang="en" hreflang="en" data-title="Multiply perfect number" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Multiplika_perfekta_nombro" title="Multiplika perfekta nombro – 世界语" lang="eo" hreflang="eo" data-title="Multiplika perfekta nombro" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_perfecto_m%C3%BAltiple" title="Número perfecto múltiple – 西班牙语" lang="es" hreflang="es" data-title="Número perfecto múltiple" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Moninkertaisesti_t%C3%A4ydellinen_luku" title="Moninkertaisesti täydellinen luku – 芬兰语" lang="fi" hreflang="fi" data-title="Moninkertaisesti täydellinen luku" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_parfait_multiple" title="Nombre parfait multiple – 法语" lang="fr" hreflang="fr" data-title="Nombre parfait multiple" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</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%A8%D7%91_%D7%9E%D7%A9%D7%95%D7%9B%D7%9C%D7%9C" title="מספר רב משוכלל – 希伯来语" lang="he" hreflang="he" data-title="מספר רב משוכלל" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%B6bbsz%C3%B6r%C3%B6sen_t%C3%B6k%C3%A9letes_sz%C3%A1mok" title="Többszörösen tökéletes számok – 匈牙利语" lang="hu" hreflang="hu" data-title="Többszörösen tökéletes számok" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_multiperfetto" title="Numero multiperfetto – 意大利语" lang="it" hreflang="it" data-title="Numero multiperfetto" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%80%8D%E7%A9%8D%E5%AE%8C%E5%85%A8%E6%95%B0" title="倍積完全数 – 日语" lang="ja" hreflang="ja" data-title="倍積完全数" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Multiperfekt_tal" title="Multiperfekt tal – 瑞典语" lang="sv" hreflang="sv" data-title="Multiperfekt tal" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%86%E0%AE%B0%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AF%86%E0%AE%B5%E0%AF%8D%E0%AE%B5%E0%AE%BF%E0%AE%AF_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="பெருக்கச் செவ்விய எண் – 泰米尔语" lang="ta" hreflang="ta" data-title="பெருக்கச் செவ்விய எண்" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" title="多重完全數 – 粤语" lang="yue" hreflang="yue" data-title="多重完全數" data-language-autonym="粵語" data-language-local-name="粤语" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1755843#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" rel="discussion" title="关于此页面的讨论[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="更改语言变体" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">不转换</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="selected ca-variants-zh mw-list-item"><a href="/zh/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" lang="zh-Hant" hreflang="zh-Hant"><span>繁體</span></a></li><li id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8"><span>阅读</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8&action=history" title="本页面的早前版本。[h]" accesskey="h"><span>查看历史</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="工具" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span 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href="/wiki/%E5%85%85%E4%BB%BD%E5%BF%85%E8%A6%81%E6%A2%9D%E4%BB%B6" class="mw-redirect" title="充份必要條件">充份必要條件</a>是<i>n</i>所有正因數的和（即<a href="/wiki/%E9%99%A4%E6%95%B8%E5%87%BD%E6%95%B8" title="除數函數">除數函數</a>，<i>σ(n)</i>）等於<i>n</i>的<i>k</i>倍，此定義下，完全數的除數函數為本身的2倍，因此是2重完全數。不論<i>k</i>的數值為何，<i>k</i>重完全數都屬於多重完全數。至2004年7月為止．已經找到k為11的多重完全數。 </p><p>可以證明： </p> <ul><li>針對一<a href="/wiki/%E8%B3%AA%E6%95%B8" class="mw-redirect" title="質數">質數</a><i>p</i>，若<i>n</i>為<i>p</i>重完全數且<i>p</i>無法整除<i>n</i>，則<i>pn</i>為(<i>p</i>+1)重完全數。因此可推得若整數<i>n</i>3重完全數，可被2整除但不能可被4整除，其充份必要條件是<i>n</i>/2需為奇數的完全數，至2012年12月為止，尚未發現任何奇數的完全數。</li> <li>若3<i>n</i>為4<i>k</i>重完全數，且3無法整除<i>n</i>，則n為3<i>k</i>-重完全數。</li></ul> <div class="mw-heading mw-heading2"><h2 id="最小的k重完全數"><span id=".E6.9C.80.E5.B0.8F.E7.9A.84k.E9.87.8D.E5.AE.8C.E5.85.A8.E6.95.B8"></span>最小的<i>k</i>重完全數</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8&action=edit&section=1" title="编辑章节：最小的k重完全數"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>以下列出<i>k</i> <= 7時，各<i>k</i>值最小的<i>k</i>重完全數（<a href="/wiki/%E6%95%B4%E6%95%B8%E6%95%B8%E5%88%97%E7%B7%9A%E4%B8%8A%E5%A4%A7%E5%85%A8" title="整數數列線上大全">OEIS</a>數列<a href="//oeis.org/A007539" class="extiw" title="oeis:A007539">A007539</a>）： </p> <table class="wikitable"> <tbody><tr> <th><i>k</i></th> <th>最小的<i>k</i>重完全數</th> <th>發現者 </th></tr> <tr> <td>1</td> <td><a href="/wiki/1" title="1">1</a></td> <td>不可考 </td></tr> <tr> <td>2</td> <td><a href="/wiki/6" title="6">6</a></td> <td>不可考 </td></tr> <tr> <td>3</td> <td><a href="/wiki/120" title="120">120</a></td> <td>不可考 </td></tr> <tr> <td>4</td> <td>30240</td> <td><a href="/wiki/%E5%8B%92%E5%86%85%C2%B7%E7%AC%9B%E5%8D%A1%E5%84%BF" class="mw-redirect" title="勒内·笛卡儿">勒内·笛卡儿</a>，約在1638年 </td></tr> <tr> <td>5</td> <td>14182439040</td> <td>勒内·笛卡儿，約在1638年 </td></tr> <tr> <td>6</td> <td>154345556085770649600</td> <td><a href="/wiki/%E7%BE%85%E4%BC%AF%E7%89%B9%C2%B7%E4%B8%B9%E5%B0%BC%C2%B7%E5%8D%A1%E9%82%81%E5%85%8B%E7%88%BE" title="羅伯特·丹尼·卡邁克爾">羅伯特·丹尼·卡邁克爾</a>, 1907 </td></tr> <tr> <td>7</td> <td>141310897947438348259849402738485523264343544818565120000</td> <td>TE Mason, 1911 </td></tr></tbody></table> <p>例如，120的除數函數滿足以下的關係： </p><p>1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = 360 = 3 × 120. </p><p>120的除數函數為120的三倍，因此為3重完全數：<br /> </p> <div class="mw-heading mw-heading2"><h2 id="參考資料"><span id=".E5.8F.83.E8.80.83.E8.B3.87.E6.96.99"></span>參考資料</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8&action=edit&section=2" title="编辑章节：參考資料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation journal">Laatsch, Richard. <a rel="nofollow" class="external text" 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Math. Mag. 2003, <b>76</b> (4): 299–301. <a rel="nofollow" class="external text" href="//www.jstor.org/stable/3219086"><span title="JSTOR">JSTOR 3219086</span></a>. <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=1573698"><span title="數學評論">MR 1573698</span></a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8&rft.atitle=A+simpler+dense+proof+regarding+the+abundancy+index&rft.aufirst=Richard+F.&rft.aulast=Ryan&rft.date=2003&rft.genre=article&rft.issue=4&rft.jtitle=Math.+Mag.&rft.pages=299-301&rft.volume=76&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1573698&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F3219086&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_mathematics-magazine_2003-10_76_4%2Fpage%2F299&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Guy, Richard K. <a rel="nofollow" class="external text" href="https://archive.org/details/unsolvedproblems0003guyr">Unsolved problems in number theory</a> 3rd. Springer-Verlag. 2004. B2. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-387-20860-2" title="Special:网络书源/978-0-387-20860-2"><span title="国际标准书号">ISBN</span> 978-0-387-20860-2</a>. <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:1058.11001"><span title="Zentralblatt MATH">Zbl 1058.11001</span></a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8&rft.aufirst=Richard+K.&rft.aulast=Guy&rft.btitle=Unsolved+problems+in+number+theory&rft.date=2004&rft.edition=3rd&rft.genre=book&rft.isbn=978-0-387-20860-2&rft.pages=B2&rft.pub=Springer-Verlag&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1058.11001&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Funsolvedproblems0003guyr&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation journal">Broughan, Kevin A.; Zhou, Qizhi. Odd multiperfect numbers of abundancy 4. J. Number Theory. 2008, <b>126</b> (6): 1566–1575. <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=2419178"><span title="數學評論">MR 2419178</span></a>. <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2007.02.001"><span title="數位物件識別號">doi:10.1016/j.jnt.2007.02.001</span></a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8&rft.atitle=Odd+multiperfect+numbers+of+abundancy+4&rft.au=Zhou%2C+Qizhi&rft.aufirst=Kevin+A.&rft.aulast=Broughan&rft.date=2008&rft.genre=article&rft.issue=6&rft.jtitle=J.+Number+Theory&rft.pages=1566-1575&rft.volume=126&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D2419178&rft_id=info%3Adoi%2F10.1016%2Fj.jnt.2007.02.001&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation arxiv">Ward, Jeffrey. Does ten have a friend?. <span class="plainlinks"><a rel="nofollow" class="external text" href="//arxiv.org/abs/0806.1001"><span title="arXiv">arXiv:0806.1001</span></a> <span typeof="mw:File"><span title="可免费查阅"><img alt="可免费查阅" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8&rft.atitle=Does+ten+have+a+friend%3F&rft.aufirst=Jeffrey&rft.aulast=Ward&rft.genre=preprint&rft.jtitle=arXiv&rft_id=info%3Aarxiv%2F0806.1001&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="外部連結"><span id=".E5.A4.96.E9.83.A8.E9.80.A3.E7.B5.90"></span>外部連結</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%A4%9A%E9%87%8D%E5%AE%8C%E5%85%A8%E6%95%B8&action=edit&section=3" title="编辑章节：外部連結"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://wwwhomes.uni-bielefeld.de/achim/mpn.html">The Multiply Perfect Numbers page</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20200219180458/http://wwwhomes.uni-bielefeld.de/achim/mpn.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a rel="nofollow" class="external text" href="http://primes.utm.edu/glossary/page.php?sort=MultiplyPerfect">The Prime Glossary: Multiply perfect numbers</a> （<a rel="nofollow" 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5em">和因數有關的整數分類</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">簡介</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%95%B4%E6%95%B0%E5%88%86%E8%A7%A3" title="整数分解">質因數分解</a></li> <li><a href="/wiki/%E5%9B%A0%E6%95%B8" title="因數">因數</a></li> <li><a href="/wiki/%E5%85%83%E5%9B%A0%E6%95%B8" title="元因數">元因數</a></li> <li><a href="/wiki/%E9%99%A4%E6%95%B8%E5%87%BD%E6%95%B8" title="除數函數">除數函數</a></li> <li><a href="/wiki/%E8%B3%AA%E5%9B%A0%E6%95%B8" title="質因數">質因數</a></li> <li><a href="/wiki/%E7%AE%97%E6%9C%AF%E5%9F%BA%E6%9C%AC%E5%AE%9A%E7%90%86" title="算术基本定理">算术基本定理</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0px 0px 0px 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/200px-Lattice_of_the_divisibility_of_60.svg.png" decoding="async" width="200" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/300px-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/400px-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%">依因數分解分類</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%B4%A8%E6%95%B0" title="质数">质数</a></li> <li><a href="/wiki/%E5%90%88%E6%95%B0" title="合数">合数</a></li> <li><a href="/wiki/%E5%8D%8A%E7%B4%A0%E6%95%B0" title="半素数">半素数</a></li> <li><a href="/wiki/%E6%99%AE%E6%B4%9B%E5%B0%BC%E5%85%8B%E6%95%B0" title="普洛尼克数">普洛尼克数</a></li> <li><a href="/wiki/%E6%A5%94%E5%BD%A2%E6%95%B0" title="楔形数">楔形数</a></li> <li><a href="/wiki/%E6%97%A0%E5%B9%B3%E6%96%B9%E5%9B%A0%E5%AD%90%E6%95%B0" title="无平方因子数">无平方数因数的数</a></li> <li><a href="/wiki/%E5%86%AA%E6%95%B8" title="冪數">冪數</a></li> <li><a href="/wiki/%E8%B3%AA%E6%95%B8%E5%86%AA" title="質數冪">質數冪</a></li> <li><a href="/wiki/%E5%B9%B3%E6%96%B9%E6%95%B8" class="mw-redirect" title="平方數">平方數</a></li> <li><a href="/wiki/%E7%AB%8B%E6%96%B9%E6%95%B8" title="立方數">立方數</a></li> <li><a href="/wiki/%E6%AC%A1%E6%96%B9%E6%95%B8" title="次方數">次方數</a></li> <li><a href="/wiki/%E9%98%BF%E5%96%80%E7%90%89%E6%96%AF%E6%95%B8" title="阿喀琉斯數">阿喀琉斯數</a></li> <li><a href="/wiki/%E5%85%89%E6%BB%91%E6%95%B8" title="光滑數">光滑數</a></li> <li><a href="/wiki/%E6%AD%A3%E8%A7%84%E6%95%B0_(%E6%95%B4%E6%95%B0)" title="正规数 (整数)">正规数</a></li> <li><a href="/wiki/%E7%B2%97%E7%B3%99%E6%95%B8" title="粗糙數">粗糙數</a></li> <li><a href="/wiki/%E4%B8%8D%E5%B0%8B%E5%B8%B8%E6%95%B8" title="不尋常數">不尋常數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">依因數和分類</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数">完全数</a></li> <li><a href="/wiki/%E6%AE%86%E5%AE%8C%E5%85%A8%E6%95%B8" title="殆完全數">殆完全數</a></li> <li><a href="/wiki/%E5%87%86%E5%AE%8C%E5%85%A8%E6%95%B0" title="准完全数">准完全数</a></li> <li><a class="mw-selflink selflink">多重完全數</a></li> <li><a href="/wiki/Hemiperfect%E6%95%B8" title="Hemiperfect數">Hemiperfect數</a></li> <li><span class="ilh-all" data-orig-title="Hyperperfect number" data-lang-code="en" data-lang-name="英语" data-foreign-title="Hyperperfect number"><span class="ilh-page"><a href="/w/index.php?title=Hyperperfect_number&action=edit&redlink=1" class="new" title="Hyperperfect number（页面不存在）">Hyperperfect number</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Hyperperfect_number" class="extiw" title="en:Hyperperfect number"><span lang="en" dir="auto">Hyperperfect number</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%B6%85%E5%AE%8C%E5%85%A8%E6%95%B8" title="超完全數">超完全數</a></li> <li><a href="/wiki/%E5%85%83%E5%AE%8C%E5%85%A8%E6%95%B8" title="元完全數">元完全數</a></li> <li><a href="/wiki/%E5%8D%8A%E5%AE%8C%E5%85%A8%E6%95%B0" title="半完全数">半完全数</a></li> <li><a href="/wiki/%E6%9C%AC%E5%8E%9F%E5%8D%8A%E5%AE%8C%E5%85%A8%E6%95%B0" title="本原半完全数">本原半完全数</a></li> <li><a href="/wiki/%E5%AF%A6%E9%9A%9B%E6%95%B8" title="實際數">實際數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">有許多因數</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%BF%87%E5%89%A9%E6%95%B0" title="过剩数">过剩数</a></li> <li><a href="/wiki/%E6%9C%AC%E5%8E%9F%E9%81%8E%E5%89%A9%E6%95%B8" title="本原過剩數">本原過剩數</a></li> <li><a href="/wiki/%E9%AB%98%E9%81%8E%E5%89%A9%E6%95%B8" title="高過剩數">高過剩數</a></li> <li><a href="/wiki/%E8%B6%85%E9%81%8E%E5%89%A9%E6%95%B8" title="超過剩數">超過剩數</a></li> <li><a href="/wiki/%E5%8F%AF%E7%BE%85%E8%96%A9%E9%87%8C%E9%81%8E%E5%89%A9%E6%95%B8" title="可羅薩里過剩數">可羅薩里過剩數</a></li> <li><a href="/wiki/%E9%AB%98%E5%90%88%E6%88%90%E6%95%B0" title="高合成数">高合成数</a></li> <li><span class="ilh-all" data-orig-title="Superior highly composite number" data-lang-code="en" data-lang-name="英语" data-foreign-title="Superior highly composite number"><span class="ilh-page"><a href="/w/index.php?title=Superior_highly_composite_number&action=edit&redlink=1" class="new" title="Superior highly composite number（页面不存在）">Superior highly composite number</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Superior_highly_composite_number" class="extiw" title="en:Superior highly composite number"><span lang="en" dir="auto">Superior highly composite number</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%A5%87%E7%95%B0%E6%95%B8_(%E6%95%B8%E8%AB%96)" title="奇異數 (數論)">奇異數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">和<a href="/wiki/%E7%9C%9F%E5%9B%A0%E5%AD%90%E5%92%8C%E6%95%B8%E5%88%97" title="真因子和數列">真因子和數列</a>有關</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%B8%8D%E5%8F%AF%E5%8F%8A%E6%95%B0" title="不可及数">不可及数</a></li> <li><a href="/wiki/%E7%9B%B8%E4%BA%B2%E6%95%B0" title="相亲数">相亲数</a></li> <li><a href="/wiki/%E7%9B%B8%E4%BA%B2%E6%95%B0%E9%93%BE" title="相亲数链">交際數</a></li> <li><a href="/wiki/%E5%A9%9A%E7%B4%84%E6%95%B8" title="婚約數">婚約數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">其他</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%BA%8F%E6%95%B0" title="亏数">亏数</a></li> <li><a href="/wiki/%E5%8F%8B%E8%AA%BC%E6%95%B8" title="友誼數">友誼數</a></li> <li><a href="/wiki/%E5%8F%8B%E8%AA%BC%E6%95%B8" title="友誼數">孤獨數</a></li> <li><a href="/wiki/%E5%8D%93%E8%B6%8A%E6%95%B0" title="卓越数">卓越数</a></li> <li><a href="/wiki/%E6%AD%90%E7%88%BE%E8%AA%BF%E5%92%8C%E6%95%B8" title="歐爾調和數">歐爾調和數</a></li> <li><a href="/wiki/%E4%BD%A9%E6%9C%8D%E6%95%B8" title="佩服數">佩服數</a></li> <li><a href="/wiki/%E7%AF%80%E5%84%89%E6%95%B8" title="節儉數">節儉數</a></li> <li><a href="/wiki/%E7%AD%89%E6%95%B8%E4%BD%8D%E6%95%B8" title="等數位數">等數位數</a></li> <li><a href="/wiki/%E5%A5%A2%E4%BE%88%E6%95%B8" title="奢侈數">奢侈數</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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