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[o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98+%ED%95%A9" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98+%ED%95%A9" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. 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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="사이트"> <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="목차" 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">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</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">처음 위치</div> </a> </li> <li id="toc-예시" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#예시"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>예시</span> </div> </a> <ul id="toc-예시-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-숫자들_집단의_특징화" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#숫자들_집단의_특징화"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>숫자들 집단의 특징화</span> </div> </a> <ul id="toc-숫자들_집단의_특징화-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-반복" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#반복"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>반복</span> </div> </a> <ul id="toc-반복-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-관련_정보" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#관련_정보"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>관련 정보</span> </div> </a> <ul id="toc-관련_정보-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-각주" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#각주"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>각주</span> </div> </a> <ul id="toc-각주-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-외부_링크" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#외부_링크"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>외부 링크</span> </div> </a> <ul id="toc-외부_링크-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="목차" 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="목차 토글" > <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">목차 토글</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">진약수의 합</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="다른 언어로 문서를 방문합니다. 11개 언어로 읽을 수 있습니다" > <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">11개 언어</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 – 영어" lang="en" hreflang="en" data-title="Aliquot sum" data-language-autonym="English" data-language-local-name="영어" 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 – 스페인어" lang="es" hreflang="es" data-title="Suma alícuota" data-language-autonym="Español" data-language-local-name="스페인어" 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 – 히브리어" lang="he" hreflang="he" data-title="סכום המנה" data-language-autonym="עברית" data-language-local-name="히브리어" 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 – 헝가리어" lang="hu" hreflang="hu" data-title="Valódiosztóösszeg-függvény" data-language-autonym="Magyar" data-language-local-name="헝가리어" 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 – 인도네시아어" lang="id" hreflang="id" data-title="Jumlah alikuot" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Aliquotsom" title="Aliquotsom – 네덜란드어" lang="nl" hreflang="nl" data-title="Aliquotsom" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" 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 – 폴란드어" lang="pl" hreflang="pl" data-title="Suma alikwotowa" data-language-autonym="Polski" data-language-local-name="폴란드어" 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 – 루마니아어" lang="ro" hreflang="ro" data-title="Sumă alicotă" data-language-autonym="Română" data-language-local-name="루마니아어" 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 – 타밀어" lang="ta" hreflang="ta" data-title="தகு வகுஎண் சார்பு" data-language-autonym="தமிழ்" data-language-local-name="타밀어" 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 – 베트남어" lang="vi" hreflang="vi" data-title="Tổng ước số thực sự" data-language-autonym="Tiếng Việt" data-language-local-name="베트남어" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%9C%9F%E5%9B%A0%E6%95%B8%E5%92%8C" title="真因數和 – 중국어" lang="zh" hreflang="zh" data-title="真因數和" data-language-autonym="中文" data-language-local-name="중국어" class="interlanguage-link-target"><span>中文</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="언어 간 링크 편집" class="wbc-editpage">링크 편집</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="이름공간"> <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/%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9" title="본문 보기 [c]" accesskey="c"><span>문서</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%ED%86%A0%EB%A1%A0:%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9&action=edit&redlink=1" rel="discussion" class="new" title="문서의 내용에 대한 토론 문서 (없는 문서) [t]" accesskey="t"><span>토론</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <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="언어 변종 바꾸기" > <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">한국어</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="보기"> <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/%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9"><span>읽기</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9&action=edit" title="이 문서의 원본 코드를 편집 [e]" accesskey="e"><span>편집</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9&action=history" title="이 문서의 과거 편집 내역입니다. [h]" accesskey="h"><span>역사 보기</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="페이지 도구"> <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="도구" > <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 class="vector-dropdown-label-text">도구</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header 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<i>n</i> )은, <i>n</i>의 자기 자신을 제외한 <i>n</i> 에 대한 모든 약수(진약수)의 합이다. 이를 수식으로 표현하면, </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 \nolimits _{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> <msub> <mo movablelimits="false">∑</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> </msub> <mi>d</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n)=\sum \nolimits _{d|n,\ d\neq n}d.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c332ea8055b4de869dc09a2bfad88b84f4d1a336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.696ex; height:4.343ex;" alt="{\displaystyle s(n)=\sum \nolimits _{d|n,\ d\neq n}d.}"></span></dd></dl> <p><a href="/wiki/%EC%86%8C%EC%88%98_(%EC%88%98%EB%A1%A0)" title="소수 (수론)">소수</a>, <a href="/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수">완전수</a>, <a href="/wiki/%EC%82%AC%EA%B5%90%EC%88%98" title="사교수">사교수</a>, <a href="/wiki/%EB%B6%80%EC%A1%B1%EC%88%98" title="부족수">부족수</a>, <a href="/wiki/%EA%B3%BC%EC%9E%89%EC%88%98" title="과잉수">과잉수</a>, <a href="/wiki/%EB%B6%88%EA%B0%80%EC%B4%89_%EC%88%98" title="불가촉 수">불가촉 수</a>를 묘사할 수 있으며, 진약수의 합 수열을 정의하는 데 사용할 수 있다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="예시"><span id=".EC.98.88.EC.8B.9C"></span>예시</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9&action=edit&section=1" title="부분 편집: 예시"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>예를 들어, 12의 진약수(즉, 12를 제외한 12의 양의 약수)는 1, 2, 3, 4, 6이므로, 12에 대한 진약수의 합은 16이다. (1 + 2 + 3 + 4 + 6). </p><p><i>n</i> = 1, 2, 3, ...에 대한 <i>s</i>(<i>n</i>)의 값은 다음과 같다. </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, ... (<a href="/wiki/%EC%98%A8%EB%9D%BC%EC%9D%B8_%EC%A0%95%EC%88%98%EC%97%B4_%EC%82%AC%EC%A0%84" title="온라인 정수열 사전">OEIS</a>의 수열 <a href="//oeis.org/A001065" class="extiw" title="oeis:A001065">A001065</a>)</dd></dl> <div class="mw-heading mw-heading2"><h2 id="숫자들_집단의_특징화"><span id=".EC.88.AB.EC.9E.90.EB.93.A4_.EC.A7.91.EB.8B.A8.EC.9D.98_.ED.8A.B9.EC.A7.95.ED.99.94"></span>숫자들 집단의 특징화</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9&action=edit&section=2" title="부분 편집: 숫자들 집단의 특징화"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>진약수의 합은 특정 숫자 집단을 묘사하는 데 사용할 수 있다. </p> <ul><li>1은 진약수의 합이 0인 유일한 수다. 어떤 숫자가 진약수의 합이 1인 경우에만 <a href="/wiki/%EC%86%8C%EC%88%98_(%EC%88%98%EB%A1%A0)" title="소수 (수론)">소수</a>다.<sup id="cite_ref-pp_1-0" class="reference"><a href="#cite_note-pp-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수">완전수</a>는 진약수의 합이 자기 자신과 같고, <a href="/wiki/%EB%B6%80%EC%A1%B1%EC%88%98" title="부족수">부족수</a>는 진약수의 합이 자기 자신보다 작으며, <a href="/wiki/%EA%B3%BC%EC%9E%89%EC%88%98" title="과잉수">과잉수</a>는 진약수의 합이 자기 자신보다 크다.<sup id="cite_ref-pp_1-1" class="reference"><a href="#cite_note-pp-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/%EC%A4%80%EC%99%84%EC%A0%84%EC%88%98" title="준완전수">준완전수</a>(이러한 수가 존재하는 경우)는 진약수의 합이 <i>n + 1</i> 과 같은 숫자 <i>n 이다</i>. 근완전수 (2의 거듭제곱 포함되며, 지금까지 알려진 유일한 숫자)는 진약수의 합이 <i>n − 1</i> 과 같은 숫자 <i>n</i>이다.<sup id="cite_ref-pp_1-2" class="reference"><a href="#cite_note-pp-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/%EB%B6%88%EA%B0%80%EC%B4%89_%EC%88%98" title="불가촉 수">불가촉 수</a>는 진약수의 합의 값으로 표현할 수 없는 수이다. 적어도 2와 5는 진약수의 합의 결과로 나올 수 없다고 관찰한 연구 Abu Mansur al-Baghdadi (서기 1000년경)도 있다.<sup id="cite_ref-pp_1-3" class="reference"><a href="#cite_note-pp-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-s_2-0" class="reference"><a href="#cite_note-s-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> 그리고 <a href="/wiki/%EC%97%90%EB%A5%B4%EB%90%98%EC%8B%9C_%ED%8C%94" title="에르되시 팔">Paul Erdős</a>는 불가촉 수가 무한하다는 것을 증명했다.<sup id="cite_ref-e_3-0" class="reference"><a href="#cite_note-e-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> 5가 진약수의 합으로 유일하게 건드릴 수 없는 홀수라는 추측은 아직 증명되지 않았지만, 준소수 <i>pq에</i> 대해 진약수의 합은 <i>p</i> + <i>q</i> + 1이라는 관찰과 함께 <a href="/wiki/%EA%B3%A8%EB%93%9C%EB%B0%94%ED%9D%90%EC%9D%98_%EC%B6%94%EC%B8%A1" class="mw-redirect" title="골드바흐의 추측">Goldbach의 추측</a> 형식에서 따를 것이다.<sup id="cite_ref-pp_1-4" class="reference"><a href="#cite_note-pp-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul> <p><a href="#CITEREFPollackPomerance2016">Pollack & Pomerance (2016)</a><span class="error harv-error" style="display: none; font-size:100%"> harvtxt error: 대상 없음: CITEREFPollackPomerance2016 (<a href="https://en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" class="extiw" title="en:Category:Harv and Sfn template errors">help</a>)</span>에서 수학자 Erdős의 "가장 좋아하는 조사 주제" 중 하나가 진약수의 합이라고 언급했다. </p> <div class="mw-heading mw-heading2"><h2 id="반복"><span id=".EB.B0.98.EB.B3.B5"></span>반복</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9&action=edit&section=3" title="부분 편집: 반복"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>음이 아닌 정수 <i>n</i> 에 대한 진약수의 합 함수를 반복하면, 진약수의 합 수열 <i>n</i>, <i>s</i> (<i>n</i>), <i>s</i>(<i>s</i>(<i>n</i>)), ...이 생성된다 (이 수열에서는 <i>s</i> (0) = 0을 정의한다). 이러한 수열이 항상 <a href="/wiki/%EC%86%8C%EC%88%98_(%EC%88%98%EB%A1%A0)" title="소수 (수론)">소수</a>로 끝나는지, <a href="/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수">완전수</a>인지, <a href="/wiki/%EC%82%AC%EA%B5%90%EC%88%98" title="사교수">사교수</a>의 주기적인 수열로 끝나는지는 알 수 없다.<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> <div class="mw-heading mw-heading2"><h2 id="관련_정보"><span id=".EA.B4.80.EB.A0.A8_.EC.A0.95.EB.B3.B4"></span>관련 정보</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9&action=edit&section=4" title="부분 편집: 관련 정보"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%EC%95%BD%EC%88%98_%ED%95%A8%EC%88%98" title="약수 함수">약수 함수</a> : 숫자의 (<i>x</i> 의 거듭제곱) 양의 약수의 합</li> <li>William of Auberive, 진약수의 합에 관심이 있는 중세시대의 수비학자</li></ul> <div class="mw-heading mw-heading2"><h2 id="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9&action=edit&section=5" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%EC%95%BD%EC%88%98_%ED%95%A8%EC%88%98" title="약수 함수">약수 함수</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="각주"><span id=".EA.B0.81.EC.A3.BC"></span>각주</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9&action=edit&section=6" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-pp-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-pp_1-0">가</a></sup> <sup><a href="#cite_ref-pp_1-1">나</a></sup> <sup><a href="#cite_ref-pp_1-2">다</a></sup> <sup><a href="#cite_ref-pp_1-3">라</a></sup> <sup><a href="#cite_ref-pp_1-4">마</a></sup></span> <span class="reference-text"><cite id="CITEREFPollackPomerance2016" class="citation">Pollack, Paul; <a href="/w/index.php?title=Carl_Pomerance&action=edit&redlink=1" class="new" title="Carl Pomerance (없는 문서)">Pomerance, Carl</a> (2016), “Some problems of Erdős on the sum-of-divisors function”, 《Transactions of the American Mathematical Society》, Series B <b>3</b>: 1–26, <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090%2Fbtran%2F10">10.1090/btran/10</a>, <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=3481968">3481968</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Some+problems+of+Erd%C5%91s+on+the+sum-of-divisors+function&rft.volume=3&rft.pages=1-26&rft.date=2016&rft_id=info%3Adoi%2F10.1090%2Fbtran%2F10&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D3481968&rft.aulast=Pollack&rft.aufirst=Paul&rft.au=Pomerance%2C+Carl&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98+%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-s-2"><span class="mw-cite-backlink"><a href="#cite_ref-s_2-0">↑</a></span> <span class="reference-text"><cite id="CITEREFSesiano1991" class="citation">Sesiano, J. (1991), “Two problems of number theory in Islamic times”, 《Archive for History of Exact Sciences》 <b>41</b> (3): 235–238, <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2FBF00348408">10.1007/BF00348408</a>, <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a> <a rel="nofollow" class="external text" href="//www.jstor.org/stable/41133889">41133889</a>, <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=1107382">1107382</a>, <a href="/wiki/%EC%8B%9C%EB%A7%A8%ED%8B%B1_%EC%8A%A4%EC%B9%BC%EB%9D%BC" title="시맨틱 스칼라">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:115235810">115235810</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=Two+problems+of+number+theory+in+Islamic+times&rft.volume=41&rft.issue=3&rft.pages=235-238&rft.date=1991&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A115235810&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.aulast=Sesiano&rft.aufirst=J.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98+%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-e-3"><span class="mw-cite-backlink"><a href="#cite_ref-e_3-0">↑</a></span> <span class="reference-text"><cite id="CITEREFErdős1973" class="citation"><a href="/w/index.php?title=Paul_Erd%C5%91s&action=edit&redlink=1" class="new" title="Paul Erdős (없는 문서)">Erdős, P.</a> (1973), <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》 <b>28</b>: 83–86, <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0337733">0337733</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Elemente+der+Mathematik&rft.atitle=%C3%9Cber+die+Zahlen+der+Form+MATH+RENDER+ERROR+und+MATH+RENDER+ERROR&rft.volume=28&rft.pages=83-86&rft.date=1973&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0337733&rft.aulast=Erd%C5%91s&rft.aufirst=P.&rft_id=https%3A%2F%2Fusers.renyi.hu%2F~p_erdos%2F1973-27.pdf&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98+%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span> <span style="font-size:100%" class="error citation-comment"><code style="color:inherit; border:inherit; padding:inherit;">|title=</code>에 지움 문자가 있음(위치 26) (<a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%9D%B8%EC%9A%A9_%EC%98%A4%EB%A5%98_%EB%8F%84%EC%9B%80%EB%A7%90#invisible_char" title="위키백과:인용 오류 도움말">도움말</a>)</span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/index.html">“Catalan's Aliquot Sequence Conjecture”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Catalan%27s+Aliquot+Sequence+Conjecture&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2Findex.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98+%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="외부_링크"><span id=".EC.99.B8.EB.B6.80_.EB.A7.81.ED.81.AC"></span>외부 링크</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9&action=edit&section=7" title="부분 편집: 외부 링크"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/RestrictedDivisorFunction.html">“Restricted Divisor Function”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Restricted+Divisor+Function&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FRestrictedDivisorFunction.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98+%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">원본 주소 "<a dir="ltr" href="https://ko.wikipedia.org/w/index.php?title=진약수의_합&oldid=38129649">https://ko.wikipedia.org/w/index.php?title=진약수의_합&oldid=38129649</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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