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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-例子" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#例子"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>例子</span> </div> </a> <ul id="toc-例子-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-數字類別的性質" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#數字類別的性質"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>數字類別的性質</span> </div> </a> <ul id="toc-數字類別的性質-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-疊代" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#疊代"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>疊代</span> </div> </a> <ul id="toc-疊代-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參見" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參見"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>參見</span> </div> </a> <ul id="toc-參見-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考文獻" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參考文獻"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>參考文獻</span> </div> </a> <ul id="toc-參考文獻-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">6</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">真因數和</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。11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-11" 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">11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-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Aliquot_sum" title="Aliquot sum – 英语" lang="en" hreflang="en" data-title="Aliquot sum" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Suma_al%C3%ADcuota" title="Suma alícuota – 西班牙语" lang="es" hreflang="es" data-title="Suma alícuota" 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-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%9B%D7%95%D7%9D_%D7%94%D7%9E%D7%A0%D7%94" title="סכום המנה – 希伯来语" lang="he" hreflang="he" data-title="סכום המנה" 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/Val%C3%B3dioszt%C3%B3%C3%B6sszeg-f%C3%BCggv%C3%A9ny" title="Valódiosztóösszeg-függvény – 匈牙利语" lang="hu" hreflang="hu" data-title="Valódiosztóösszeg-függvény" 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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Jumlah_alikuot" title="Jumlah alikuot – 印度尼西亚语" lang="id" hreflang="id" data-title="Jumlah alikuot" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9" title="진약수의 합 – 韩语" lang="ko" hreflang="ko" data-title="진약수의 합" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Aliquotsom" title="Aliquotsom – 荷兰语" lang="nl" hreflang="nl" data-title="Aliquotsom" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Suma_alikwotowa" title="Suma alikwotowa – 波兰语" lang="pl" hreflang="pl" data-title="Suma alikwotowa" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sum%C4%83_alicot%C4%83" title="Sumă alicotă – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Sumă alicotă" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%95%E0%AF%81_%E0%AE%B5%E0%AE%95%E0%AF%81%E0%AE%8E%E0%AE%A3%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81" title="தகு வகுஎண் சார்பு – 泰米尔语" lang="ta" hreflang="ta" data-title="தகு வகுஎண் சார்பு" 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-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%E1%BB%95ng_%C6%B0%E1%BB%9Bc_s%E1%BB%91_th%E1%BB%B1c_s%E1%BB%B1" title="Tổng ước số thực sự – 越南语" lang="vi" hreflang="vi" data-title="Tổng ước số thực sự" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</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/Q22912133#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/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C" 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:%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C" 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/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C" 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/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C" 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/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C" 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/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C" 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/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C" 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/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C" 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/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C" 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/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/60px-Information_icon4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/80px-Information_icon4.svg.png 2x" data-file-width="620" data-file-height="620" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">由于已知的技术原因，图表暂时不可用。带来不便，我们深表歉意。<span class="hide-when-compact"></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> </div></div></div><div class="thumbcaption">真因數和<a href="/wiki/%E5%87%BD%E6%95%B8%E5%9C%96%E5%BD%A2" class="mw-redirect" title="函數圖形">函數的圖形</a></div></div></div> <p><b>真因數和</b>，又稱<b>真因子和</b>，在<a href="/wiki/%E6%95%B8%E8%AB%96" class="mw-redirect" title="數論">數論</a>中，一個正<a href="/wiki/%E6%95%B4%E6%95%B8" class="mw-redirect" title="整數">整數</a>的所有<a href="/wiki/%E7%9C%9F%E5%9B%A0%E6%95%B8" class="mw-redirect" title="真因數">真因數</a>之和，即除了自己本身外的所有正<a href="/wiki/%E5%9B%A0%E6%95%B8" title="因數">因數</a>之和，通常以<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d659eab6d8874e2cb6124fd2fc66b477b128c006" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.294ex; height:2.843ex;" alt="{\displaystyle s(n)}"></span>來表示： </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 s(n)=\sum _{d|n,\ d\neq n}d.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo>,</mo> <mtext> </mtext> <mi>d</mi> <mo>≠</mo> <mi>n</mi> </mrow> </munder> <mi>d</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n)=\sum _{d|n,\ d\neq n}d.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94df39e6789cf7f8eb29871fe42bd6acc950659f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:16.109ex; height:6.009ex;" alt="{\displaystyle s(n)=\sum _{d|n,\ d\neq n}d.}"></span></dd></dl> <p>真因數和可以用來描述<a href="/wiki/%E8%B3%AA%E6%95%B8" class="mw-redirect" title="質數">質數</a>、<a href="/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数">完全数</a>、<a href="/wiki/%E7%9B%B8%E8%A6%AA%E6%95%B8%E9%8F%88" class="mw-redirect" title="相親數鏈">相親數鏈</a>、<a href="/wiki/%E4%BA%8F%E6%95%B0" title="亏数">亏数</a>、<a href="/wiki/%E8%BF%87%E5%89%A9%E6%95%B0" title="过剩数">过剩数</a>和<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>，也可以用於定義整數的<a href="/wiki/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C%E6%95%B8%E5%88%97" class="mw-redirect" title="真因數和數列">真因數和數列</a>。 </p><p>真因數和函數<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d659eab6d8874e2cb6124fd2fc66b477b128c006" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.294ex; height:2.843ex;" alt="{\displaystyle s(n)}"></span>與1次<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><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 _{1}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e6dcdcf20087da1d219fdc818382bcc77f4fbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.586ex; height:2.843ex;" alt="{\displaystyle \sigma _{1}(n)}"></span>的關係僅差<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>：<sup id="cite_ref-mathworld_2-0" class="reference"><a href="#cite_note-mathworld-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(n)=\sigma _{1}(n)-n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n)=\sigma _{1}(n)-n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8fd60df7a93cd8757c44cf9f2e137a5d21ab0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.214ex; height:2.843ex;" alt="{\displaystyle s(n)=\sigma _{1}(n)-n}"></span></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="例子"><span id=".E4.BE.8B.E5.AD.90"></span>例子</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C&action=edit&section=1" title="编辑章节：例子"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>以12為例，12的<a href="/wiki/%E7%9C%9F%E5%9B%A0%E6%95%B8" class="mw-redirect" title="真因數">真因數</a>（即除了自己本身外的所有正因數）有1、2、3、4和6，則其真因數和為<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}+{2}}+{3}}+{4}}+{6}}=16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mrow> <mo>=</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{{{{1}+{2}}+{3}}+{4}}+{6}}=16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/299577d37b9fc848398985d4f2fbafc48f3de672" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.597ex; height:2.343ex;" alt="{\displaystyle {{{{{1}+{2}}+{3}}+{4}}+{6}}=16}"></span> </p><p>下面數列呈現前幾個整數的真因數和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(n),n=1,2,3,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n),n=1,2,3,...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf3c164a46573a9303005e43c18e7e352ce3d168" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.125ex; height:2.843ex;" alt="{\displaystyle s(n),n=1,2,3,...}"></span><sup id="cite_ref-mathworld_2-1" class="reference"><a href="#cite_note-mathworld-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>0、 1、 1、 3、 1、 6、 1、 7、 4、 8、 1、 16、 1、 10、 9、 15、 1、 21、 1、 22、 11、 14、 1、 36、 6、 16、 13、 28、 1、 42、 1、 31、 15、 20、 13、 55、 1、 22、 17、 50、 1、 54、 1、 40、 33、 26、 1、 76、 8、 43、 21 …… （<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/A001065" class="extiw" title="oeis:A001065">A001065</a>）</dd></dl> <div class="mw-heading mw-heading2"><h2 id="數字類別的性質"><span id=".E6.95.B8.E5.AD.97.E9.A1.9E.E5.88.A5.E7.9A.84.E6.80.A7.E8.B3.AA"></span>數字類別的性質</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C&action=edit&section=2" title="编辑章节：數字類別的性質"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>真因數和函數可以用來區分幾個特別的數字類別： </p> <ul><li>1是唯一一個真因數和為0的正整數。</li> <li>如果一個正整數真因數和為1則代表該數是一個<a href="/wiki/%E8%B3%AA%E6%95%B8" class="mw-redirect" title="質數">質數</a><sup id="cite_ref-pp_3-0" class="reference"><a href="#cite_note-pp-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。</li> <li><a href="/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数">完全数</a>的真因數和等於本身、<a href="/wiki/%E4%BA%8F%E6%95%B0" title="亏数">亏数</a>的真因數和小於本身、<a href="/wiki/%E8%BF%87%E5%89%A9%E6%95%B0" title="过剩数">过剩数</a>的真因數和大於本身<sup id="cite_ref-pp_3-1" class="reference"><a href="#cite_note-pp-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。<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>（如果存在的話）真因數和為<i>n+1</i>。<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>（目前已知僅有2的<a href="/wiki/%E5%86%AA" title="冪">冪</a>）真因數和為<i>n-1</i>。</li> <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>是指不是任何數之真因數和的數。相關研究至少可以追溯到大約公元1000年<span class="ilh-all" data-orig-title="伊本·塔希爾·巴格達迪" data-lang-code="en" data-lang-name="英语" data-foreign-title="Abu Mansur al-Baghdadi"><span class="ilh-page"><a href="/w/index.php?title=%E4%BC%8A%E6%9C%AC%C2%B7%E5%A1%94%E5%B8%8C%E7%88%BE%C2%B7%E5%B7%B4%E6%A0%BC%E9%81%94%E8%BF%AA&action=edit&redlink=1" class="new" title="伊本·塔希爾·巴格達迪（页面不存在）">伊本·塔希爾·巴格達迪</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/Abu_Mansur_al-Baghdadi" class="extiw" title="en:Abu Mansur al-Baghdadi"><span lang="en" dir="auto">Abu Mansur al-Baghdadi</span></a></span>）</span></span>的研究，其發現2和5都是不可及數<sup id="cite_ref-pp_3-2" class="reference"><a href="#cite_note-pp-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-s_4-0" class="reference"><a href="#cite_note-s-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>。<a href="/wiki/%E5%9F%83%E5%B0%94%E5%BE%B7%E4%BB%80%C2%B7%E5%B8%95%E5%B0%94" title="埃尔德什·帕尔">埃尔德什·帕尔</a>證明有無限多個不可及數<sup id="cite_ref-e_5-0" class="reference"><a href="#cite_note-e-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>。目前尚未確定5是否為唯一的奇數不可及數，但可以從<a href="/wiki/%E5%93%A5%E5%BE%B7%E5%B7%B4%E8%B5%AB%E7%8C%9C%E6%83%B3" title="哥德巴赫猜想">哥德巴赫猜想</a>的一種形式與<a href="/wiki/%E5%8D%8A%E7%B4%A0%E6%95%B0" title="半素数">半素数</a><i>pq</i>的真因數和為<i>p+q+1</i>的觀察得出<sup id="cite_ref-pp_3-3" class="reference"><a href="#cite_note-pp-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。</li></ul> <p>數學家保羅·波拉克（Paul Pollack）和<span class="ilh-all ilh-blue" data-orig-title="卡爾·帕梅朗斯" data-lang-code="en" data-lang-name="英语" data-foreign-title="Carl Pomerance"><span class="ilh-page"><a href="/wiki/%E5%8D%A1%E7%88%BE%C2%B7%E5%B8%95%E6%A2%85%E6%9C%97%E6%96%AF" title="卡爾·帕梅朗斯">卡爾·帕梅朗斯</a></span></span>指出，<a href="/wiki/%E5%9F%83%E5%B0%94%E5%BE%B7%E4%BB%80%C2%B7%E5%B8%95%E5%B0%94" title="埃尔德什·帕尔">埃尔德什·帕尔</a>「最喜歡的研究項目」是真因數和。<sup id="cite_ref-pp_3-4" class="reference"><a href="#cite_note-pp-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="疊代"><span id=".E7.96.8A.E4.BB.A3"></span>疊代</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C&action=edit&section=3" title="编辑章节：疊代"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r84833064">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">主条目：<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></div> <p><a href="/wiki/%E8%BF%AD%E4%BB%A3%E5%87%BD%E6%95%B0" title="迭代函数">疊代</a>真因數和函數可以產生非負整數的<a href="/wiki/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C%E6%95%B8%E5%88%97" class="mw-redirect" title="真因數和數列">真因數和數列</a><style data-mw-deduplicate="TemplateStyles:r58896141">.mw-parser-output .serif{font-family:Times,serif}</style><span class="serif"><span class="texhtml"><i>n</i>, <i>s</i>(<i>n</i>), <i>s</i>(<i>s</i>(<i>n</i>)), ...</span></span>（在這個數列中，我們定義<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml"><i>s</i>(0) = 0</span></span>）。 </p><p><a href="/wiki/%E7%9B%B8%E8%A6%AA%E6%95%B8%E9%8F%88" class="mw-redirect" title="相親數鏈">相親數鏈</a>的<a href="/wiki/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C%E6%95%B8%E5%88%97" class="mw-redirect" title="真因數和數列">真因數和數列</a>為<span class="ilh-all" data-orig-title="週期數列" data-lang-code="en" data-lang-name="英语" data-foreign-title="Periodic sequence"><span class="ilh-page"><a href="/wiki/%E5%91%A8%E6%9C%9F%E6%95%B0%E5%88%97" class="mw-redirect" title="周期数列">週期數列</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/Periodic_sequence" class="extiw" title="en:Periodic sequence"><span lang="en" dir="auto">Periodic sequence</span></a></span>）</span></span>，<a href="/wiki/%E7%9B%B8%E8%A6%AA%E6%95%B8" class="mw-redirect" title="相親數">相親數</a>是週期為2的相親數鏈。 </p><p>目前尚不清楚這些數列是否總是以<a href="/wiki/%E8%B3%AA%E6%95%B8" class="mw-redirect" title="質數">質數</a>、<a href="/wiki/%E5%AE%8C%E5%85%A8%E6%95%B0" title="完全数">完全数</a>或周期性的<a href="/wiki/%E7%9B%B8%E4%BA%B2%E6%95%B0%E9%93%BE" title="相亲数链">相亲数链</a>為結尾。<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="參見"><span id=".E5.8F.83.E8.A6.8B"></span>參見</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C&action=edit&section=4" title="编辑章节：參見"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><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>：一數之正因數的<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml"><i>x</i></span></span>次方和</li> <li><span class="ilh-all" data-orig-title="紀堯姆·德奧貝里夫" data-lang-code="en" data-lang-name="英语" data-foreign-title="William of Auberive"><span class="ilh-page"><a href="/w/index.php?title=%E7%B4%80%E5%A0%AF%E5%A7%86%C2%B7%E5%BE%B7%E5%A5%A7%E8%B2%9D%E9%87%8C%E5%A4%AB&action=edit&redlink=1" class="new" title="紀堯姆·德奧貝里夫（页面不存在）">紀堯姆·德奧貝里夫</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/William_of_Auberive" class="extiw" title="en:William of Auberive"><span lang="en" dir="auto">William of Auberive</span></a></span>）</span></span>：中世紀的命理學家，對真因數和感興趣</li></ul> <div class="mw-heading mw-heading2"><h2 id="參考文獻"><span id=".E5.8F.83.E8.80.83.E6.96.87.E7.8D.BB"></span>參考文獻</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C&action=edit&section=5" title="编辑章节：參考文獻"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-mathworld-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-mathworld_2-0"><sup><b>1.0</b></sup></a> <a href="#cite_ref-mathworld_2-1"><sup><b>1.1</b></sup></a></span> <span class="reference-text"><cite class="citation web"><a href="/wiki/%E5%9F%83%E9%87%8C%E5%85%8B%C2%B7%E9%9F%A6%E6%96%AF%E5%9D%A6%E5%9B%A0" title="埃里克·韦斯坦因">Weisstein, Eric W.</a> (编). <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/RestrictedDivisorFunction.html">Restricted Divisor Function</a>. at <a href="/wiki/MathWorld" title="MathWorld"><i>MathWorld</i></a>--A Wolfram Web Resource. <a href="/wiki/%E6%B2%83%E5%B0%94%E5%A4%AB%E5%8B%92%E5%A7%86%E7%A0%94%E7%A9%B6%E5%85%AC%E5%8F%B8" title="沃尔夫勒姆研究公司">Wolfram Research, Inc.</a> <span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="连接到英语网页">（英语）</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C&rft.atitle=Restricted+Divisor+Function&rft.aufirst=Eric+W.&rft.aulast=Weisstein&rft.genre=unknown&rft.jtitle=at+%27%27MathWorld%27%27--A+Wolfram+Web+Resource&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FRestrictedDivisorFunction.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-pp-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-pp_3-0"><sup><b>2.0</b></sup></a> <a href="#cite_ref-pp_3-1"><sup><b>2.1</b></sup></a> <a href="#cite_ref-pp_3-2"><sup><b>2.2</b></sup></a> <a href="#cite_ref-pp_3-3"><sup><b>2.3</b></sup></a> <a href="#cite_ref-pp_3-4"><sup><b>2.4</b></sup></a></span> <span class="reference-text"><cite id="CITEREFPollackPomerance2016" class="citation">Pollack, Paul; Pomerance, Carl, Some problems of Erdős on the sum-of-divisors function, Transactions of the American Mathematical Society, Series B, 2016, <b>3</b>: 1–26, <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=3481968"><span title="數學評論">MR 3481968</span></a>, <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fbtran%2F10"><span title="數位物件識別號">doi:10.1090/btran/10</span></a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C&rft.atitle=Some+problems+of+Erd%C5%91s+on+the+sum-of-divisors+function&rft.au=Pomerance%2C+Carl&rft.aufirst=Paul&rft.aulast=Pollack&rft.date=2016&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.pages=1-26&rft.volume=3&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D3481968&rft_id=info%3Adoi%2F10.1090%2Fbtran%2F10&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-s-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-s_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFSesiano1991" class="citation">Sesiano, J., Two problems of number theory in Islamic times, Archive for History of Exact Sciences, 1991, <b>41</b> (3): 235–238, <a rel="nofollow" class="external text" href="//www.jstor.org/stable/41133889"><span title="JSTOR">JSTOR 41133889</span></a>, <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=1107382"><span title="數學評論">MR 1107382</span></a>, <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00348408"><span title="數位物件識別號">doi:10.1007/BF00348408</span></a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C&rft.atitle=Two+problems+of+number+theory+in+Islamic+times&rft.aufirst=J.&rft.aulast=Sesiano&rft.date=1991&rft.genre=article&rft.issue=3&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.pages=235-238&rft.volume=41&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1107382&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F41133889&rft_id=info%3Adoi%2F10.1007%2FBF00348408&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-e-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-e_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFErdős1973" class="citation"><a href="/wiki/Paul_Erd%C5%91s" class="mw-redirect" title="Paul Erdős">Erdős, P.</a>, <a rel="nofollow" class="external text" href="https://users.renyi.hu/~p_erdos/1973-27.pdf">Über die Zahlen der Form <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}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (n)-n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6126e22ff95eb6b21e5259913dc29a0969f1d7f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.769ex; height:2.843ex;" alt="{\displaystyle \sigma (n)-n}"></span> und <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-\phi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-\phi (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e68375697ca97eb71949c5b6656c86aba33681d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.825ex; height:2.843ex;" alt="{\displaystyle n-\phi (n)}"></span></a> <span style="font-size:85%;">(PDF)</span>, Elemente der Mathematik, 1973, <b>28</b>: 83–86 <span class="reference-accessdate"> [<span class="nowrap">2022-09-22</span>]</span>, <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0337733"><span title="數學評論">MR 0337733</span></a>, （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20220805230426/https://users.renyi.hu/~p_erdos/1973-27.pdf">存档</a> <span style="font-size:85%;">(PDF)</span>于2022-08-05）</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C&rft.atitle=%C3%9Cber+die+Zahlen+der+Form+MATH+RENDER+ERROR+und+MATH+RENDER+ERROR&rft.aufirst=P.&rft.aulast=Erd%C5%91s&rft.date=1973&rft.genre=article&rft.jtitle=Elemente+der+Mathematik&rft.pages=83-86&rft.volume=28&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0337733&rft_id=https%3A%2F%2Fusers.renyi.hu%2F~p_erdos%2F1973-27.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite class="citation web"><a href="/wiki/%E5%9F%83%E9%87%8C%E5%85%8B%C2%B7%E9%9F%A6%E6%96%AF%E5%9D%A6%E5%9B%A0" title="埃里克·韦斯坦因">Weisstein, Eric W.</a> (编). <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/CatalansAliquotSequenceConjecture.html">Catalan's Aliquot Sequence Conjecture</a>. at <a href="/wiki/MathWorld" title="MathWorld"><i>MathWorld</i></a>--A Wolfram Web Resource. <a href="/wiki/%E6%B2%83%E5%B0%94%E5%A4%AB%E5%8B%92%E5%A7%86%E7%A0%94%E7%A9%B6%E5%85%AC%E5%8F%B8" title="沃尔夫勒姆研究公司">Wolfram Research, Inc.</a> <span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="连接到英语网页">（英语）</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C&rft.atitle=Catalan%27s+Aliquot+Sequence+Conjecture&rft.aufirst=Eric+W.&rft.aulast=Weisstein&rft.genre=unknown&rft.jtitle=at+%27%27MathWorld%27%27--A+Wolfram+Web+Resource&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FCatalansAliquotSequenceConjecture.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <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=%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C&action=edit&section=6" title="编辑章节：外部連結"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Restricted_Divisor_Function"><cite class="citation web"><a href="/wiki/%E5%9F%83%E9%87%8C%E5%85%8B%C2%B7%E9%9F%A6%E6%96%AF%E5%9D%A6%E5%9B%A0" title="埃里克·韦斯坦因">埃里克·韦斯坦因</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/RestrictedDivisorFunction.html">Restricted Divisor Function</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span 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