[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki13\/archives\/306510#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki13\/archives\/306510","headline":"\u5b8c\u5168\u30c8\u30fc\u30c6\u30a3\u30a8\u30f3\u30c8\u6570 – Wikipedia","name":"\u5b8c\u5168\u30c8\u30fc\u30c6\u30a3\u30a8\u30f3\u30c8\u6570 – Wikipedia","description":"\u5b8c\u5168\u30c8\u30fc\u30c6\u30a3\u30a8\u30f3\u30c8\u6570\uff08\u304b\u3093\u305c\u3093\u30c8\u30fc\u30c6\u30a3\u30a8\u30f3\u30c8\u3059\u3046\u3001\u82f1: perfect totient number\uff09\u3001\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u306f\u3001\u81ea\u7136\u6570\u306e\u3046\u3061\u3001\u4ee5\u4e0b\u306e\u7b49\u5f0f\u3092\u6e80\u305f\u3059\u6570 n \u3067\u3042\u308b\u3002 n=\u2211i=1c+1\u03c6i(n)=\u03c6(n)+\u03c6(\u03c6(n))+\u03c6(\u03c6(\u03c6(n)))+\u22ef+\u03c6(\u03c6(\u22ef(\u03c6(\u03c6\u23dec+1(n)))\u22ef)){displaystyle n=sum _{i=1}^{c+1}varphi ^{i}(n)=varphi (n)+varphi (varphi (n))+varphi (varphi (varphi (n)))+cdots +overbrace {varphi (varphi (cdots","datePublished":"2022-04-20","dateModified":"2022-04-20","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki13\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki13\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0e800eff5306ceda7390e8c7dd2861201a40be08","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0e800eff5306ceda7390e8c7dd2861201a40be08","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki13\/archives\/306510","about":["Wiki"],"wordCount":1274,"articleBody":"\u5b8c\u5168\u30c8\u30fc\u30c6\u30a3\u30a8\u30f3\u30c8\u6570\uff08\u304b\u3093\u305c\u3093\u30c8\u30fc\u30c6\u30a3\u30a8\u30f3\u30c8\u3059\u3046\u3001\u82f1: perfect totient number\uff09\u3001\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u306f\u3001\u81ea\u7136\u6570\u306e\u3046\u3061\u3001\u4ee5\u4e0b\u306e\u7b49\u5f0f\u3092\u6e80\u305f\u3059\u6570 n \u3067\u3042\u308b\u3002n=\u2211i=1c+1\u03c6i(n)=\u03c6(n)+\u03c6(\u03c6(n))+\u03c6(\u03c6(\u03c6(n)))+\u22ef+\u03c6(\u03c6(\u22ef(\u03c6(\u03c6\u23dec+1(n)))\u22ef)){displaystyle n=sum _{i=1}^{c+1}varphi ^{i}(n)=varphi (n)+varphi (varphi (n))+varphi (varphi (varphi (n)))+cdots +overbrace {varphi (varphi (cdots (varphi (varphi } ^{c+1}(n)))cdots ))}\u03c6i(n)={\u03c6(n)i=1\u03c6(\u03c6i\u22121(n))i\u22652{displaystyle varphi ^{i}(n)=left{{begin{matrix}varphi (n)qquad i=1\\varphi (varphi ^{i-1}(n))quad igeq 2end{matrix}}right.}\u3053\u3053\u3067 \u03c6 \u306f\u30aa\u30a4\u30e9\u30fc\u306e\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u95a2\u6570\u3067\u3042\u308b\u3002\u4f8b\u3048\u3070 327 \u306f\u03c6(327) = 216, \u03c6(216) = 72, \u03c6(72) = 24, \u03c6(24) = 8, \u03c6(8) = 4, \u03c6(4) = 2, \u03c6(2) = 1\u3068 1 \u306b\u306a\u308b\u307e\u3067\u6b21\u3005\u3068 \u03c6 \u95a2\u6570\u306e\u5024\u3092\u8a08\u7b97\u3057\u3001\u305d\u308c\u3089\u306e\u7dcf\u548c\u304c 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327 \u3068\u5143\u306e\u6570\u306b\u7b49\u3057\u304f\u306a\u308b\u306e\u3067\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u3067\u3042\u308b\u3002\u4e00\u822c\u306b\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570 n \u306f\u4ee5\u4e0b\u306e\u5f0f\u3092\u6e80\u305f\u3059\u3002\u03c6c(n)=2{displaystyle displaystyle varphi ^{c}(n)=2}\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u306f\u7121\u6570\u306b\u3042\u308a\u3001\u305d\u306e\u3046\u3061\u6700\u5c0f\u306e\u6570\u306f 3 \u3067\u3042\u308b\u3002\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u3092\u5c0f\u3055\u3044\u9806\u306b\u5217\u8a18\u3059\u308b\u30683, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, … \uff08\u30aa\u30f3\u30e9\u30a4\u30f3\u6574\u6570\u5217\u5927\u8f9e\u5178\u306e\u6570\u5217 A082897\uff09\u307b\u3068\u3093\u3069\u306e\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u306f 3 \u306e\u500d\u6570\u3067\u3042\u308a\u30013 \u306e\u500d\u6570\u3067\u306a\u3044\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u306e\u3046\u3061\u6700\u5c0f\u306e\u6570\u306f 4375 \u3067\u3042\u308b\u3002\u7279\u306b 3 \u306e\u7d2f\u4e57\u6570 (3, 9, 27, 81, 243, 729, 2187, \u2026) \u306f\u5168\u3066\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u3067\u3042\u308b\u3002\u3053\u308c\u306f 3 \u306e\u7d2f\u4e57\u6570 3k \u304c\u03c6(3k)=\u03c6(2\u00d73k)=2\u00d73k\u22121{displaystyle displaystyle varphi (3^{k})=varphi (2times 3^{k})=2times 3^{k-1}}\u3092\u6e80\u305f\u3059\u3053\u3068\u304b\u3089\u8a3c\u660e\u3067\u304d\u308b\u3002Venkataraman \u306f1975\u5e74\u306b\u7d20\u6570 p \u304c p = 4\u00d73k + 1 \u306e\u5f62\u3067\u8868\u3055\u308c\u308b\u3068\u304d\u30013p \u304c\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u306b\u306a\u308b\u3053\u3068\u3092\u767a\u898b\u3057\u305f\u3002\u4e00\u822c\u306b\u3001\u7d20\u6570 p > 3 \u306b\u5bfe\u3057\u3066 3p \u304c\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u3067\u3042\u308b\u3068\u304d\u3001p\u22611 (mod 4) \u3067\u3042\u308b (Mohan, Suryanarayana 1982)\u3002\u3057\u304b\u3057\u3001\u3053\u306e\u5f62\u3092\u3057\u305f 3p \u306e\u5168\u3066\u304c\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u306b\u306a\u308b\u8a33\u3067\u306f\u306a\u3044\u3002\u4f8b\u3048\u3070 p = 17 \u306e\u5834\u5408 p\u22611 (mod 4) \u3092\u6e80\u305f\u3057\u30013p = 51 \u3068\u306a\u308b\u304c\u300151 \u306f\u5b8c\u5168\u30c8\u30fc\u30b7\u30a7\u30f3\u30c8\u6570\u3067\u306f\u306a\u3044\u3002\u95a2\u9023\u4e8b\u9805[\u7de8\u96c6]\u3053\u306e\u8a18\u4e8b\u306f\u3001\u30af\u30ea\u30a8\u30a4\u30c6\u30a3\u30d6\u30fb\u30b3\u30e2\u30f3\u30ba\u30fb\u30e9\u30a4\u30bb\u30f3\u30b9 \u8868\u793a-\u7d99\u627f 3.0 \u975e\u79fb\u690d\u306e\u3082\u3068\u63d0\u4f9b\u3055\u308c\u3066\u3044\u308b\u30aa\u30f3\u30e9\u30a4\u30f3\u6570\u5b66\u8f9e\u5178\u300ePlanetMath\u300f\u306e\u9805\u76eeperfect totient number\u306e\u672c\u6587\u3092\u542b\u3080"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki13\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki13\/archives\/306510#breadcrumbitem","name":"\u5b8c\u5168\u30c8\u30fc\u30c6\u30a3\u30a8\u30f3\u30c8\u6570 – Wikipedia"}}]}]