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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="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=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=%EA%B3%BC%EC%9E%89%EC%88%98" 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=%EA%B3%BC%EC%9E%89%EC%88%98" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</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/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</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="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-과잉수의_예" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#과잉수의_예"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>과잉수의 예</span> </div> </a> <ul id="toc-과잉수의_예-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-관련_성질" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#관련_성질"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>관련 성질</span> </div> </a> <ul id="toc-관련_성질-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-기타" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#기타"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>기타</span> </div> </a> <ul id="toc-기타-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">4</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-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" title="목차" > <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">과잉수</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. 43개 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-43" 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">43개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%B2%D8%A7%D8%A6%D8%AF" title="عدد زائد – 아랍어" lang="ar" hreflang="ar" data-title="عدد زائد" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%97%D0%B0%D0%BB%D1%96%D1%88%D0%BD%D1%96%D1%8F_%D0%BB%D1%96%D0%BA%D1%96" title="Залішнія лікі – 벨라루스어" lang="be" hreflang="be" data-title="Залішнія лікі" data-language-autonym="Беларуская" data-language-local-name="벨라루스어" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%91%D0%BE%D0%B3%D0%B0%D1%82%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Богато число – 불가리아어" lang="bg" hreflang="bg" data-title="Богато число" data-language-autonym="Български" data-language-local-name="불가리아어" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%AE%E0%A7%83%E0%A6%A6%E0%A7%8D%E0%A6%A7_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="সমৃদ্ধ সংখ্যা – 벵골어" lang="bn" hreflang="bn" data-title="সমৃদ্ধ সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="벵골어" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Niver_puilh" title="Niver puilh – 브르타뉴어" lang="br" hreflang="br" data-title="Niver puilh" data-language-autonym="Brezhoneg" data-language-local-name="브르타뉴어" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_abundant" title="Nombre abundant – 카탈로니아어" lang="ca" hreflang="ca" data-title="Nombre abundant" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B2%DB%8C%D8%A7%D8%AF%DB%95_%DA%98%D9%85%D8%A7%D8%B1%DB%95" title="زیادە ژمارە – 소라니 쿠르드어" lang="ckb" hreflang="ckb" data-title="زیادە ژمارە" data-language-autonym="کوردی" data-language-local-name="소라니 쿠르드어" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Abundantn%C3%AD_%C4%8D%C3%ADslo" title="Abundantní číslo – 체코어" lang="cs" hreflang="cs" data-title="Abundantní číslo" data-language-autonym="Čeština" data-language-local-name="체코어" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Excessivt_tal" title="Excessivt tal – 덴마크어" lang="da" hreflang="da" data-title="Excessivt tal" data-language-autonym="Dansk" data-language-local-name="덴마크어" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Abundante_Zahl" title="Abundante Zahl – 독일어" lang="de" hreflang="de" data-title="Abundante Zahl" data-language-autonym="Deutsch" data-language-local-name="독일어" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A5%CF%80%CE%B5%CF%81%CF%84%CE%AD%CE%BB%CE%B5%CE%B9%CE%BF%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Υπερτέλειος αριθμός – 그리스어" lang="el" hreflang="el" data-title="Υπερτέλειος αριθμός" data-language-autonym="Ελληνικά" data-language-local-name="그리스어" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B9mer_abund%C3%A0nt" title="Nùmer abundànt – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nùmer abundànt" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Abundant_number" title="Abundant number – 영어" lang="en" hreflang="en" data-title="Abundant number" data-language-autonym="English" data-language-local-name="영어" 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/Abunda_nombro" title="Abunda nombro – 에스페란토어" lang="eo" hreflang="eo" data-title="Abunda nombro" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" 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_abundante" title="Número abundante – 스페인어" lang="es" hreflang="es" data-title="Número abundante" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_oparo" title="Zenbaki oparo – 바스크어" lang="eu" hreflang="eu" data-title="Zenbaki oparo" data-language-autonym="Euskara" data-language-local-name="바스크어" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D8%B2%D8%A7%D8%A6%D8%AF" title="اعداد زائد – 페르시아어" lang="fa" hreflang="fa" data-title="اعداد زائد" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Runsas_luku" title="Runsas luku – 핀란드어" lang="fi" hreflang="fi" data-title="Runsas luku" data-language-autonym="Suomi" data-language-local-name="핀란드어" 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_abondant" title="Nombre abondant – 프랑스어" lang="fr" hreflang="fr" data-title="Nombre abondant" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_abundante" title="Número abundante – 갈리시아어" lang="gl" hreflang="gl" data-title="Número abundante" data-language-autonym="Galego" data-language-local-name="갈리시아어" class="interlanguage-link-target"><span>Galego</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%A9%D7%95%D7%A4%D7%A2" title="מספר שופע – 히브리어" lang="he" hreflang="he" data-title="מספר שופע" 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/B%C5%91velked%C5%91_sz%C3%A1mok" title="Bővelkedő számok – 헝가리어" lang="hu" hreflang="hu" data-title="Bővelkedő számok" data-language-autonym="Magyar" data-language-local-name="헝가리어" 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_abbondante" title="Numero abbondante – 이탈리아어" lang="it" hreflang="it" data-title="Numero abbondante" data-language-autonym="Italiano" data-language-local-name="이탈리아어" 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/%E9%81%8E%E5%89%B0%E6%95%B0" title="過剰数 – 일본어" lang="ja" hreflang="ja" data-title="過剰数" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_abundans" title="Numerus abundans – 라틴어" lang="la" hreflang="la" data-title="Numerus abundans" data-language-autonym="Latina" data-language-local-name="라틴어" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_bondant" title="Numer bondant – 롬바르드어" lang="lmo" hreflang="lmo" data-title="Numer bondant" data-language-autonym="Lombard" data-language-local-name="롬바르드어" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Overvloedig_getal" title="Overvloedig getal – 네덜란드어" lang="nl" hreflang="nl" data-title="Overvloedig getal" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Overskuddstall" title="Overskuddstall – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Overskuddstall" data-language-autonym="Norsk bokmål" data-language-local-name="노르웨이어(보크말)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_nadmiarowe" title="Liczby nadmiarowe – 폴란드어" lang="pl" hreflang="pl" data-title="Liczby nadmiarowe" data-language-autonym="Polski" data-language-local-name="폴란드어" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_abundante" title="Número abundante – 포르투갈어" lang="pt" hreflang="pt" data-title="Número abundante" data-language-autonym="Português" data-language-local-name="포르투갈어" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_abundent" title="Număr abundent – 루마니아어" lang="ro" hreflang="ro" data-title="Număr abundent" data-language-autonym="Română" data-language-local-name="루마니아어" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%B7%D0%B1%D1%8B%D1%82%D0%BE%D1%87%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Избыточное число – 러시아어" lang="ru" hreflang="ru" data-title="Избыточное число" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/N%C3%B9mmuru_abbunnanti" title="Nùmmuru abbunnanti – 시칠리아어" lang="scn" hreflang="scn" data-title="Nùmmuru abbunnanti" data-language-autonym="Sicilianu" data-language-local-name="시칠리아어" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Abundant_number" title="Abundant number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Abundant number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Abundantn%C3%A9_%C4%8D%C3%ADslo" title="Abundantné číslo – 슬로바키아어" lang="sk" hreflang="sk" data-title="Abundantné číslo" data-language-autonym="Slovenčina" data-language-local-name="슬로바키아어" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Obilno_%C5%A1tevilo" title="Obilno število – 슬로베니아어" lang="sl" hreflang="sl" data-title="Obilno število" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ymnigt_tal" title="Ymnigt tal – 스웨덴어" lang="sv" hreflang="sv" data-title="Ymnigt tal" data-language-autonym="Svenska" data-language-local-name="스웨덴어" 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%AE%E0%AE%BF%E0%AE%95%E0%AF%88%E0%AE%AF%E0%AF%86%E0%AE%A3%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="மிகையெண் (கணிதம்) – 타밀어" lang="ta" hreflang="ta" 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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">보이기</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">숨기기</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">위키백과, 우리 모두의 백과사전.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(<a href="/w/index.php?title=%ED%99%80%EC%88%98_%EA%B3%BC%EC%9E%89%EC%88%98&redirect=no" class="mw-redirect" title="홀수 과잉수">홀수 과잉수</a>에서 넘어옴)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ko" dir="ltr"><p><span class="nowrap"></span> <a href="/wiki/%EC%88%98%EB%A1%A0" title="수론">수론</a>에서 <b>과잉수</b>(過剩數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">abundant number</span>)는 <a href="/wiki/%EC%A7%84%EC%95%BD%EC%88%98%EC%9D%98_%ED%95%A9" title="진약수의 합">진약수의 합</a>(자기 자신을 제외한 양의 <a href="/wiki/%EC%95%BD%EC%88%98" title="약수">약수</a>를 모두 더한 값)이 자기 자신보다 큰 <a href="/wiki/%EC%9E%90%EC%97%B0%EC%88%98" title="자연수">자연수</a>다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="과잉수의_예"><span id=".EA.B3.BC.EC.9E.89.EC.88.98.EC.9D.98_.EC.98.88"></span>과잉수의 예</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EA%B3%BC%EC%9E%89%EC%88%98&action=edit&section=1" title="부분 편집: 과잉수의 예"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>예를 들어, 20의 <a href="/wiki/%EC%A7%84%EC%95%BD%EC%88%98" class="mw-redirect" title="진약수">진약수</a>의 합은 <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+4+5+10=22>20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>5</mn> <mo>+</mo> <mn>10</mn> <mo>=</mo> <mn>22</mn> <mo>></mo> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+2+4+5+10=22>20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb42c3321a49f2a8cbf5538397e79d45df530e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:29.183ex; height:2.343ex;" alt="{\displaystyle 1+2+4+5+10=22>20}"></span>으로 원래의 수 20보다 더 크기 때문에 20은 과잉수가 된다. </p><p>과잉수는 무수히 많이 있으며, 과잉수 중 가장 작은 수는 <a href="/wiki/12" title="12">12</a>이다. 특이하게도 이는 진약수의 일부의 합에 따라 <a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수</a>와 <a href="/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수">기묘수</a>로 분류할 수 있다고 한다. <a href="/wiki/%EC%A7%84%EC%95%BD%EC%88%98" class="mw-redirect" title="진약수">진약수</a>의 일부의 합으로 자기자신이 된다면 ‘<a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수</a>’(semiperfect nunber), 그렇지 못하면 즉 과잉수 중에서 <a href="/wiki/%EC%A7%84%EC%95%BD%EC%88%98" class="mw-redirect" title="진약수">진약수</a>, 일부의 합으로도 자기자신과 똑같아 만들어낼 수 없으면 ‘<a href="/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수">기묘수</a>’(weird number)이다. 사실 과잉수 중 <a href="/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수">기묘수</a> 즉 다시 말해 <a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수</a>가 아닌 과잉수를 찾는 게 더 어렵다고 한다. 1부터 1만까지의 자연수 중에서 과잉수는 모두 2491개가 있는데, 이들중에서도 <a href="/wiki/%EA%B4%B4%EC%A7%9C%EC%88%98" class="mw-redirect" title="괴짜수">기묘수</a>는 단지 7개 밖에 없기 때문이다. 가장 작은 기묘수는 <a href="/wiki/70" title="70">70</a>이다. 70의 약수는 1, 2, 5, 7, 10, 14, 35, 70으로 70을 제외한 약수의 합은 74이다. </p><p>하지만 위의 약수들 중 어떤 조합을 빼고 합하더라도 70이 되도록 만들 수 있는 방법은 없다. 5 이상의 수는 아예 제외할 수 없고, 나머지는 1, 2뿐이너서 두가지 모두를 제외시켜도 71이기 때문에 그렇다. 또한 진약수가 모두 <a href="/wiki/%EB%B6%80%EC%A1%B1%EC%88%98" title="부족수">부족수</a>이어서 다른 과잉수 또는 <a href="/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수">완전수</a>의 <b><a href="/wiki/%ED%95%AD%EB%93%B1%EC%9B%90" title="항등원">자기자신이 아닌</a></b> <a href="/wiki/%EB%B0%B0%EC%88%98" class="mw-disambig" title="배수">배수</a>로 표기될 수 없는 과잉수는 <a href="/w/index.php?title=%EC%9B%90%EC%8B%9C_%EA%B3%BC%EC%9E%89%EC%88%98&action=edit&redlink=1" class="new" title="원시 과잉수 (없는 문서)">원시 과잉수</a> (primitive abundant number) 라고 한다. 마찬가지로, 다른 <a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수</a>의 <b><a href="/wiki/%ED%95%AD%EB%93%B1%EC%9B%90" title="항등원">자기자신이 아닌</a></b> <a href="/wiki/%EB%B0%B0%EC%88%98" class="mw-disambig" title="배수">배수</a>로 표기될 수 없지만, 다시 말해 진약수가 모두 부족수이거나 부족수와 <a href="/wiki/%EA%B4%B4%EC%A7%9C%EC%88%98" class="mw-redirect" title="괴짜수">괴짜수</a>가 섞여있고, 반완전수가 전혀 없더라도 특정 과잉수의 진약수들의 일부의 합이 자기자신이 될 경우 해당 <a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수</a>는 <a href="/w/index.php?title=%EC%9B%90%EC%8B%9C_%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98&action=edit&redlink=1" class="new" title="원시 반완전수 (없는 문서)">원시 반완전수</a>(primitive semi perfect number)이다. </p><p>또한 진약수가 모두 부족수여서 다른 <a href="/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수">기묘수</a>의 배수의 형식으로 표기될 수 없는 <a href="/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수">기묘수</a>는 <a href="/w/index.php?title=%EC%9B%90%EC%8B%9C_%EA%B8%B0%EB%AC%98%EC%88%98&action=edit&redlink=1" class="new" title="원시 기묘수 (없는 문서)">원시 기묘수</a>(primitive weird number)이라고 하며, 1000000 이하의 수 중 단지 24개 존재한다. 그러나 n이 <a href="/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수">기묘수</a>인 경우 n의 약수의 합보다 더 큰 소수 p와 n의 곱 역시 <a href="/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수">기묘수</a>가 된다는 성질 때문에 10000 이상에서는 갑자기 149 이상의 <a href="/wiki/%EC%86%8C%EC%88%98_(%EC%88%98%EB%A1%A0)" title="소수 (수론)">소수(素數)</a>와 최소의 <a href="/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수">기묘수</a> <a href="/wiki/70" title="70">70</a>의 곱으로 인해 기묘수가 나타나는 빈도가 높아진다. </p><p>200보다 작은 과잉수는 다음과 같다. (<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/A005101" class="extiw" title="oeis:A005101">A005101</a>) </p> <dl><dd><a href="/wiki/12" title="12">12</a>, <a href="/wiki/18" title="18">18</a>, <a href="/wiki/20" title="20">20</a>, <a href="/wiki/24" title="24">24</a>, <a href="/wiki/30" title="30">30</a>, <a href="/wiki/36" title="36">36</a>, <a href="/wiki/40" title="40">40</a>, <a href="/wiki/42" title="42">42</a>, <a href="/wiki/48" title="48">48</a>, <a href="/wiki/54" title="54">54</a>, <a href="/wiki/56" title="56">56</a>, <a href="/wiki/60" title="60">60</a>, <a href="/wiki/66" title="66">66</a>, <a href="/wiki/70" title="70">70</a>, <a href="/wiki/72" title="72">72</a>, <a href="/wiki/78" title="78">78</a>, <a href="/wiki/80" title="80">80</a>, <a href="/wiki/84" title="84">84</a>, <a href="/wiki/88" title="88">88</a>, <a href="/wiki/90" title="90">90</a>, <a href="/wiki/96" title="96">96</a>, <a href="/wiki/100" title="100">100</a>, <a href="/wiki/102" title="102">102</a>, <a href="/wiki/104" title="104">104</a>, <a href="/wiki/108" title="108">108</a>, <a href="/wiki/112" title="112">112</a>, <a href="/wiki/114" title="114">114</a>, <a href="/wiki/120" title="120">120</a>, <a href="/wiki/126" title="126">126</a>, <a href="/wiki/132" title="132">132</a>, <a href="/wiki/138" title="138">138</a>, <a href="/wiki/140" title="140">140</a>, <a href="/wiki/144" title="144">144</a>, <a href="/wiki/150" title="150">150</a>, <a href="/wiki/156" title="156">156</a>, <a href="/wiki/160" title="160">160</a>, <a href="/wiki/162" title="162">162</a>, <a href="/wiki/168" title="168">168</a>, <a href="/wiki/174" title="174">174</a>, <a href="/wiki/176" title="176">176</a>, <a href="/wiki/180" title="180">180</a>, <a href="/wiki/186" title="186">186</a>, <a href="/wiki/192" title="192">192</a>, <a href="/wiki/196" title="196">196</a>, <a href="/wiki/198" title="198">198</a>, …</dd></dl> <div class="mw-heading mw-heading2"><h2 id="관련_성질"><span id=".EA.B4.80.EB.A0.A8_.EC.84.B1.EC.A7.88"></span>관련 성질</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EA%B3%BC%EC%9E%89%EC%88%98&action=edit&section=2" title="부분 편집: 관련 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="anchor" id="제곱_인수가_없는_과잉수"></span> <b>제곱 인수가 없는 과잉수</b> (squarefree abundant number) (<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/A087248" class="extiw" title="oeis:A087248">A087248</a>)</li></ul> <dl><dd>1000보다 작은 과잉수 중 소인수에 거듭제곱이 없는 수는 다음과 같다. <dl><dd><a href="/wiki/30" title="30">30</a>, <a href="/wiki/42" title="42">42</a>, <a href="/wiki/66" title="66">66</a>, <a href="/wiki/70" title="70">70</a>, <a href="/wiki/78" title="78">78</a>, <a href="/wiki/102" title="102">102</a>, <a href="/wiki/114" title="114">114</a>, <a href="/wiki/138" title="138">138</a>, <a href="/wiki/174" title="174">174</a>, <a href="/wiki/186" title="186">186</a>, <a href="/wiki/210" title="210">210</a>, <a href="/wiki/222" title="222">222</a>, <a href="/wiki/246" title="246">246</a>, <a href="/wiki/258" title="258">258</a>, <a href="/wiki/282" title="282">282</a>, <a href="/wiki/318" title="318">318</a>, <a href="/wiki/330" title="330">330</a>, <a href="/wiki/354" title="354">354</a>, <a href="/wiki/366" title="366">366</a>, <a href="/wiki/390" title="390">390</a>, <a href="/wiki/402" title="402">402</a>, <a href="/wiki/426" title="426">426</a>, <a href="/wiki/438" title="438">438</a>, <a href="/wiki/462" title="462">462</a>, <a href="/wiki/474" title="474">474</a>, <a href="/wiki/498" title="498">498</a>, <a href="/wiki/510" title="510">510</a>, <a href="/wiki/534" title="534">534</a>, <a href="/wiki/546" title="546">546</a>, <a href="/wiki/570" title="570">570</a>, <a href="/wiki/582" title="582">582</a>, <a href="/wiki/606" title="606">606</a>, <a href="/wiki/618" title="618">618</a>, <a href="/wiki/642" title="642">642</a>, <a href="/wiki/654" title="654">654</a>, <a href="/wiki/678" title="678">678</a>, <a href="/wiki/690" title="690">690</a>, <a href="/wiki/714" title="714">714</a>, <a href="/wiki/762" title="762">762</a>, <a href="/wiki/770" title="770">770</a>, <a href="/wiki/786" title="786">786</a>, <a href="/wiki/798" title="798">798</a>, <a href="/wiki/822" title="822">822</a>, <a href="/wiki/834" title="834">834</a>, <a href="/wiki/858" title="858">858</a>, <a href="/wiki/870" title="870">870</a>, <a href="/wiki/894" title="894">894</a>, <a href="/wiki/906" title="906">906</a>, <a href="/wiki/910" title="910">910</a>, <a href="/wiki/930" title="930">930</a>, <a href="/wiki/942" title="942">942</a>, <a href="/wiki/966" title="966">966</a>, <a href="/wiki/978" title="978">978</a>, …</dd></dl></dd></dl> <ul><li><span class="anchor" id="홀수_과잉수"></span> <b>홀수 과잉수</b> (odd abundant number) (<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/A005231" class="extiw" title="oeis:A005231">A005231</a>)</li></ul> <dl><dd>과잉수 중 가장 작은 홀수는 <b><a href="/wiki/945" title="945">945</a></b>이며, 945부터 5355까지는 서로 630만큼의 등차수열을 이룬다. (즉 5775보다 작은 모든 홀수 과잉수는 315의 배수다.) 과잉수의 <a href="/wiki/%EB%B0%B0%EC%88%98" class="mw-disambig" title="배수">배수</a>는 언제나 과잉수이므로 짝수 과잉수도 홀수 과잉수도 무수히 많이 있다.</dd> <dd>2만보다 작은 홀수 과잉수는 다음과 같다. <dl><dd><a href="/wiki/945" title="945">945</a>, <a href="/wiki/1575" class="mw-redirect" title="1575">1575</a>, <a href="/wiki/2205" title="2205">2205</a>, <a href="/wiki/2835" class="mw-redirect" title="2835">2835</a>, <a href="/wiki/3465" class="mw-redirect" title="3465">3465</a>, <a href="/wiki/4095" class="mw-redirect" title="4095">4095</a>, <a href="/wiki/4725" class="mw-redirect" title="4725">4725</a>, <a href="/wiki/5355" class="mw-redirect" title="5355">5355</a>, <a href="/wiki/5775" class="mw-redirect" title="5775">5775</a>, <a href="/wiki/5985" title="5985">5985</a>, <a href="/wiki/6435" class="mw-redirect" title="6435">6435</a>, <a href="/wiki/6615" class="mw-redirect" title="6615">6615</a>, <a href="/wiki/6825" class="mw-redirect" title="6825">6825</a>, <a href="/wiki/7245" class="mw-redirect" title="7245">7245</a>, <a href="/wiki/7425" class="mw-redirect" title="7425">7425</a>, <a href="/wiki/7875" class="mw-redirect" title="7875">7875</a>, <a href="/wiki/8085" class="mw-redirect" title="8085">8085</a>, <a href="/wiki/8415" class="mw-redirect" title="8415">8415</a>, <a href="/wiki/8505" class="mw-redirect" title="8505">8505</a>, <a href="/wiki/8925" class="mw-redirect" title="8925">8925</a>, <a href="/wiki/9135" class="mw-redirect" title="9135">9135</a>, <a href="/wiki/9555" class="mw-redirect" title="9555">9555</a>, <a href="/wiki/9765" class="mw-redirect" title="9765">9765</a>, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175, 14805, 15015, 15435, 16065, 16695, 17325, 17955, 18585, 19215, 19305, 19635, 19845, …</dd></dl></dd></dl> <ul><li><span class="anchor" id="제곱_인수가_없는_홀수_과잉수"></span> <b>제곱 인수가 없는 홀수 과잉수</b> (odd squarefree abundant number) (<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/A112643" class="extiw" title="oeis:A112643">A112643</a>)</li></ul> <dl><dd>10만보다 작은 홀수 과잉수 중 소인수에 거듭제곱이 없는 수는 다음과 같다. <dl><dd>15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, …</dd></dl></dd></dl> <ul><li><b>10과 서로소인 과잉수</b> (= 5와 서로소인 홀수 과잉수) (<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/A064001" class="extiw" title="oeis:A064001">A064001</a>)</li></ul> <dl><dd>100만보다 작은 과잉수 중 10과 서로소인 수는 다음과 같다. (10과 서로소인 수는 2나 5로 나누어떨어지지 않는다.) <dl><dd>81081, 153153, 171171, 189189, 207207, 223839, 243243, 261261, 279279, 297297, 351351, 459459, 513513, 567567, 621621, 671517, 729729, 742203, 783783, 793611, 812889, 837837, 891891, 908523, 960687, 999999, …</dd></dl></dd></dl> <ul><li><b>제곱 인수가 없고 10과 서로소인 과잉수</b> (<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/A112644" class="extiw" title="oeis:A112644">A112644</a>)</li></ul> <dl><dd>소인수에 거듭제곱이 없고 10과 서로소인 가장 작은 과잉수는 <b>22,309,287</b><small>(2230만 9287)</small>이다.</dd></dl> <ul><li><b>5, 7과 서로소인 홀수 과잉수</b> (<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/A112640" class="extiw" title="oeis:A112640">A112640</a>)</li></ul> <dl><dd>홀수 과잉수 중 5로도 7로도 나누어떨어지지 않는 가장 작은 수는 <b>28,683,369</b><small>(2868만 3369)</small>다.</dd></dl> <ul><li><b>6과 서로소인 과잉수</b> (= 3과 서로소인 홀수 과잉수) (<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/A115414" class="extiw" title="oeis:A115414">A115414</a>)</li></ul> <dl><dd>6과 서로소인 가장 작은 과잉수는 <b>5,391,411,025</b><small>(53억 9141만 1025)</small>다. (6과 서로소인 수는 2나 3으로 나누어떨어지지 않는다.)</dd></dl> <div class="mw-heading mw-heading2"><h2 id="기타"><span id=".EA.B8.B0.ED.83.80"></span>기타</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EA%B3%BC%EC%9E%89%EC%88%98&action=edit&section=3" title="부분 편집: 기타"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>진약수의 일부를 더해서 자기자신이 될 수 있으면 <a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수</a>(semiperfect nunber)이고, 그렇지 못하면 <a href="/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수">기묘수</a>(weird number)가 된다. <a href="/wiki/945" title="945">945</a>는 가장 작은 홀수 반완전수이기도 하다.</li> <li>자연수 중 과잉수의 <a href="/wiki/%EC%A0%90%EA%B7%BC_%EB%B0%80%EB%8F%84" title="점근 밀도">점근 밀도</a>(Asymptotic Density)는 평균 0.2474에서 0.2480 사이로 알려져 있으므로 아무리 많아도 26%를 넘지 않는다. 그래도 과잉수의 비율이 1/6, 즉 약 16.67% 이하로 떨어지지 않음을 보이는 것은 쉬운데, 6을 제외한 6의 배수는 모두 과잉수이므로 실제로 n=30가 되었을 때 17%이 되며, n=56일 때부터 17%를 초과하게 되면서 더 이상 17% 이하로 떨어지지 않는다는 것을 알 수 있다.</li> <li>20161보다 큰 모든 정수는 2개의 과잉수의 합으로 표현될 수 있다. (2개의 과잉수의 합으로 표현되지 않는 자연수는 모두 1456개이다.)</li> <li>어떤 수의 진약수의 합이 그 어떤 수보다 1만큼 커지는 과잉수를 <a href="/wiki/%EC%A4%80%EC%99%84%EC%A0%84%EC%88%98" title="준완전수">준완전수</a>라고 한다. 이 준완전수는 지금까지 하나도 발견되지 않았다. 또한 준완전수가 존재하지 않는다는 명제도 아직까지 증명되지 않았다.(사이먼 싱 저 페르마의 마지막 정리 한국어 번역본 33쪽) 간혹, 어떤 자연수의 진약수의 합이 자기 자신과 비슷할 경우 그 자연수를 준완전수라고 칭하기도 한다. 진약수의 합에서 자기자신을 뺀 값을 <a href="/w/index.php?title=%EC%B4%88%EA%B3%BC%EA%B0%92&action=edit&redlink=1" class="new" title="초과값 (없는 문서)">초과값</a>이라고 하는데, 그 결과가 음수이면 <a href="/wiki/%EB%B6%80%EC%A1%B1%EC%88%98" title="부족수">부족수</a>, 0이면 <a href="/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수">완전수</a>, 양수이면 과잉수이며, 과잉수의 경우 제외할 일부 진약수의 합이 초과값과 일치해야 <a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수</a>가 된다. <a href="/w/index.php?title=%EC%B4%88%EA%B3%BC%EA%B0%92&action=edit&redlink=1" class="new" title="초과값 (없는 문서)">초과값</a>이 홀수이려면 짝수제곱이거나, 짝수제곱에 2의 거듭제곱을 곱해야 하기 때문이다. 3 이상의 홀수를 약수로 가지고, 진약수가 모두 홀수인 자연수는 초과값이 -1이 아니다. 참고로 n이 완전수일 경우에는 n의 소인수의 지수+1만큼의 거듭제곱이나 n과 서로소인 임의의 소수 p에 대하여 pn의 <a href="/w/index.php?title=%EC%B4%88%EA%B3%BC%EA%B0%92&action=edit&redlink=1" class="new" title="초과값 (없는 문서)">초과값</a>은 p의 값에 상관없이 항상 n×2이므로 초과값이 <a href="/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수">완전수</a>의 2배인 과잉수는 무수히 많다.</li> <li>과잉수는 모두 <a href="/wiki/%ED%95%A9%EC%84%B1%EC%88%98" title="합성수">합성수</a>이며, 모든 소수는 진약수가 1 밖에 없어서 부족수다. 그러나 합성수이면서 소인수가 2개 이상, 즉 합성수 중 소수의 거듭제곱수가 아닌 부족수도 아주 많이 있으며, 두 소수의 곱으로 나타낼 수 있는 임의의 자연수 n에 대하여 (서로 같은 소수예도 상관없다) 약수가 n개인 과잉수는 개수가 한정되어 있다. 또한 세 개 이상의 소수의 곱으로 나타낼 수 있는 임의의 자연수 n애 대하여 약수가 n개인 과잉수는 무수히 많이 있다. 특이한 점은 약수가 4개인 과잉수는 없고, 약수가 <a href="/wiki/%EC%86%8C%EC%88%98_(%EC%88%98%EB%A1%A0)" title="소수 (수론)">소수</a>개인 과잉수는 <a href="/wiki/%EC%86%8C%EC%9D%B8%EC%88%98%EB%B6%84%ED%95%B4" title="소인수분해">소인수</a>가 단 하나밖에 없기 때문에 그러헌 과잉수는 존재하자 않는다. 완전수도 자기자신을 제외한 모든 배수가 과잉수인데, <a href="/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수">완전수</a> 6의 경우 6n(n은 2 이상인 자연수)의 진약수의 합 중에서 1+n+2n+3n만 해도 6n+1로, 6n보다 크기 때문이다. 또한 완전수의 배수는 모두 반완전수인데, 그 이유는 완전수에 곱한 수만큼의 완전수의 진약수들의 합이 그 수 자신이 되기 때문이다. 예를 들어, 6n의 진약수 중에서 n, 2n, 3n을 더하기만 하면 바로 6n이 되기 때문에 <a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수</a>다.</li> <li>진약수의 합이 자기 자신보다 3이 더 큰 수는 18밖에 없다.</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=%EA%B3%BC%EC%9E%89%EC%88%98&action=edit&section=4" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수">완전수</a></li> <li><a href="/wiki/%EB%B6%80%EC%A1%B1%EC%88%98" title="부족수">부족수</a></li> <li><a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수</a></li> <li><a href="/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수">기묘수</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r36480591">.mw-parser-output .hlist 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href="/w/index.php?title=%ED%8B%80%ED%86%A0%EB%A1%A0:%EC%95%BD%EC%88%98%EC%97%90_%EB%94%B0%EB%A5%B8_%EC%A0%95%EC%88%98%EC%9D%98_%EC%A7%91%ED%95%A9&action=edit&redlink=1" class="new" title="틀토론:약수에 따른 정수의 집합 (없는 문서)"><abbr title="이 틀에 관해 토론하기" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">t</abbr></a></li><li class="nv-편집"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%AC%B8%EC%84%9C%ED%8E%B8%EC%A7%91/%ED%8B%80:%EC%95%BD%EC%88%98%EC%97%90_%EB%94%B0%EB%A5%B8_%EC%A0%95%EC%88%98%EC%9D%98_%EC%A7%91%ED%95%A9" title="특수:문서편집/틀:약수에 따른 정수의 집합"><abbr title="이 틀을 편집하기" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">e</abbr></a></li></ul></div><div id="약수에_따른_정수의_집합" style="font-size:114%;margin:0 4em">약수에 따른 정수의 집합</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">개요</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%86%8C%EC%9D%B8%EC%88%98%EB%B6%84%ED%95%B4" title="소인수분해">소인수분해</a></li> <li><a href="/wiki/%EC%95%BD%EC%88%98" title="약수">약수</a></li> <li><a href="/wiki/%EC%9C%A0%EB%8B%88%ED%83%80%EB%A6%AC_%EC%95%BD%EC%88%98" title="유니타리 약수">유니타리 약수</a></li> <li><a href="/wiki/%EC%95%BD%EC%88%98_%ED%95%A8%EC%88%98" title="약수 함수">약수 함수</a></li> <li><a href="/wiki/%EC%86%8C%EC%9D%B8%EC%88%98" class="mw-redirect" title="소인수">소인수</a></li> <li><a href="/wiki/%EC%82%B0%EC%88%A0%EC%9D%98_%EA%B8%B0%EB%B3%B8_%EC%A0%95%EB%A6%AC" title="산술의 기본 정리">산술의 기본 정리</a></li> <li><a href="/w/index.php?title=%EC%82%B0%EC%88%A0%EC%88%98&action=edit&redlink=1" class="new" title="산술수 (없는 문서)">산술수</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Lattice_of_the_divisibility_of_60.svg" class="mw-file-description" title="Divisibility of 60"><img alt="Divisibility of 60" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/175px-Lattice_of_the_divisibility_of_60.svg.png" decoding="async" width="175" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/263px-Lattice_of_the_divisibility_of_60.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/350px-Lattice_of_the_divisibility_of_60.svg.png 2x" data-file-width="313" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">인수분해 관련</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%86%8C%EC%88%98_(%EC%88%98%EB%A1%A0)" title="소수 (수론)">소수</a></li> <li><a href="/wiki/%ED%95%A9%EC%84%B1%EC%88%98" title="합성수">합성수</a></li> <li><a href="/wiki/%EB%B0%98%EC%86%8C%EC%88%98" title="반소수">반소수</a></li> <li><a href="/w/index.php?title=Pronic_number&action=edit&redlink=1" class="new" title="Pronic number (없는 문서)">Pronic</a></li> <li><a href="/w/index.php?title=%EC%90%90%EA%B8%B0%EC%88%98&action=edit&redlink=1" class="new" title="쐐기수 (없는 문서)">쐐기수</a></li> <li><a href="/wiki/%EC%A0%9C%EA%B3%B1_%EC%9D%B8%EC%88%98%EA%B0%80_%EC%97%86%EB%8A%94_%EC%A0%95%EC%88%98" title="제곱 인수가 없는 정수">제곱 인수가 없는 정수</a></li> <li><a href="/w/index.php?title=%EC%B5%9C%EA%B0%95%EB%A0%A5%EC%88%98&action=edit&redlink=1" class="new" title="최강력수 (없는 문서)">최강력수</a></li> <li><a href="/wiki/%EA%B1%B0%EB%93%AD%EC%A0%9C%EA%B3%B1%EC%88%98" class="mw-redirect" title="거듭제곱수">거듭제곱수</a></li> <li><a href="/w/index.php?title=Achilles_number&action=edit&redlink=1" class="new" title="Achilles number (없는 문서)">Achilles</a></li> <li><a href="/w/index.php?title=Smooth_number&action=edit&redlink=1" class="new" title="Smooth number (없는 문서)">Smooth</a></li> <li><a href="/w/index.php?title=Regular_number&action=edit&redlink=1" class="new" title="Regular number (없는 문서)">Regular</a></li> <li><a href="/w/index.php?title=Rough_number&action=edit&redlink=1" class="new" title="Rough number (없는 문서)">Rough</a></li> <li><a href="/w/index.php?title=Unusual_number&action=edit&redlink=1" class="new" title="Unusual number (없는 문서)">Unusual</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">약수합 관련</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%99%84%EC%A0%84%EC%88%98" title="완전수">완전수</a></li> <li><a href="/w/index.php?title=%EA%B1%B0%EC%9D%98_%EC%99%84%EC%A0%84%EC%88%98&action=edit&redlink=1" class="new" title="거의 완전수 (없는 문서)">거의(almost) 완전수</a></li> <li><a href="/wiki/%EC%A4%80%EC%99%84%EC%A0%84%EC%88%98" title="준완전수">준완전수(quasi)</a></li> <li><a href="/w/index.php?title=%EA%B3%B1%EC%99%84%EC%A0%84%EC%88%98&action=edit&redlink=1" class="new" title="곱완전수 (없는 문서)">곱완전수</a></li> <li><a href="/w/index.php?title=%EC%A0%88%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98&action=edit&redlink=1" class="new" title="절반완전수 (없는 문서)">절반완전수(Hemi)</a></li> <li><a href="/wiki/%EC%B4%88%EC%99%84%EC%A0%84%EC%88%98" title="초완전수">초완전수(hyper)</a></li> <li><a href="/w/index.php?title=%EC%9A%B0%EC%99%84%EC%A0%84%EC%88%98&action=edit&redlink=1" class="new" title="우완전수 (없는 문서)">우완전수(super)</a></li> <li><a href="/w/index.php?title=%EC%9C%A0%EB%8B%88%ED%83%80%EB%A6%AC_%EC%99%84%EC%A0%84%EC%88%98&action=edit&redlink=1" class="new" title="유니타리 완전수 (없는 문서)">유니타리 완전수</a></li> <li><a href="/wiki/%EB%B0%98%EC%99%84%EC%A0%84%EC%88%98" title="반완전수">반완전수(Semi)</a></li> <li><a href="/w/index.php?title=Practical_number&action=edit&redlink=1" class="new" title="Practical number (없는 문서)">Practical</a></li> <li><a href="/w/index.php?title=Erd%C5%91s%E2%80%93Nicolas_number&action=edit&redlink=1" class="new" title="Erdős–Nicolas number (없는 문서)">Erdős–Nicolas</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">약수가 많음</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">과잉수</a></li> <li><a href="/w/index.php?title=%EC%9B%90%EC%8B%9C_%EA%B3%BC%EC%9E%89%EC%88%98&action=edit&redlink=1" class="new" title="원시 과잉수 (없는 문서)">원시 과잉수</a></li> <li><a href="/w/index.php?title=%EA%B3%A0%EA%B3%BC%EC%9E%89%EC%88%98&action=edit&redlink=1" class="new" title="고과잉수 (없는 문서)">고과잉수</a></li> <li><a href="/w/index.php?title=%EC%B4%88%EA%B3%BC%EC%9E%89%EC%88%98&action=edit&redlink=1" class="new" title="초과잉수 (없는 문서)">초과잉수</a></li> <li><a href="/w/index.php?title=%EA%B1%B0%EB%8C%80%EA%B3%BC%EC%9E%89%EC%88%98&action=edit&redlink=1" class="new" title="거대과잉수 (없는 문서)">거대과잉수</a></li> <li><a href="/wiki/%EA%B3%A0%EB%8F%84_%ED%95%A9%EC%84%B1%EC%88%98" title="고도 합성수">고도 합성수</a></li> <li><a href="/w/index.php?title=%EC%9A%B0%EC%9B%94%EA%B3%A0%ED%95%A9%EC%84%B1%EC%88%98&action=edit&redlink=1" class="new" title="우월고합성수 (없는 문서)">우월고합성수</a></li> <li><a href="/wiki/%EA%B8%B0%EB%AC%98%EC%88%98" title="기묘수">기묘수</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/w/index.php?title=%EC%A7%84%EC%95%BD%EC%88%98%ED%95%A9_%EC%88%98%EC%97%B4&action=edit&redlink=1" class="new" title="진약수합 수열 (없는 문서)">진약수합</a> 관련</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EB%B6%88%EA%B0%80%EC%B4%89%EC%88%98" class="mw-redirect" title="불가촉수">불가촉수</a></li> <li><a href="/wiki/%EC%B9%9C%ED%99%94%EC%88%98" title="친화수">친화수</a></li> <li><a href="/wiki/%EC%82%AC%EA%B5%90%EC%88%98" title="사교수">사교수</a></li> <li><a href="/wiki/%ED%98%BC%EC%95%BD%EC%88%98" title="혼약수">혼약수</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">기타</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EB%B6%80%EC%A1%B1%EC%88%98" title="부족수">부족수</a></li> <li><a href="/wiki/%EC%B9%9C%EA%B5%AC%EC%88%98" title="친구수">친구수</a></li> <li><a href="/wiki/%EC%B9%9C%EA%B5%AC%EC%88%98#고독수" title="친구수">고독수</a></li> <li><a href="/w/index.php?title=%EC%88%AD%EA%B3%A0%EC%88%98&action=edit&redlink=1" class="new" title="숭고수 (없는 문서)">숭고수</a></li> <li><a href="/w/index.php?title=Harmonic_divisor_number&action=edit&redlink=1" class="new" title="Harmonic divisor number (없는 문서)">Harmonic divisor</a></li> <li><a href="/w/index.php?title=Frugal_number&action=edit&redlink=1" class="new" title="Frugal number (없는 문서)">Frugal</a></li> <li><a href="/w/index.php?title=Equidigital_number&action=edit&redlink=1" class="new" title="Equidigital number (없는 문서)">Equidigital</a></li> <li><a href="/w/index.php?title=Extravagant_number&action=edit&redlink=1" class="new" title="Extravagant number (없는 문서)">Extravagant</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" 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=과잉수&oldid=38536559#홀수_과잉수">https://ko.wikipedia.org/w/index.php?title=과잉수&oldid=38536559#홀수_과잉수</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%B6%84%EB%A5%98" title="특수:분류">분류</a>: <ul><li><a href="/wiki/%EB%B6%84%EB%A5%98:%EC%88%98%EB%A1%A0" title="분류:수론">수론</a></li><li><a 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