\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(n\\) \u306e\u3059\u3079\u3066\u306e\u6b63\u306e\u7d04\u6570\uff08 \\(1\\) \u3068 \\(n\\) \u3092\u542b\u3080\uff09\u306e\u548c\u3092 \\(S(n)\\) \u3068\u304a\u304f. \u4f8b\u3048\u3070, \\(S(9) = 1 +3 +9 = 13\\) \u3067\u3042\u308b. \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(n\\) \u304c\u7570\u306a\u308b\u7d20\u6570 \\(p\\) \u3068 \\(q\\) \u306b\u3088\u3063\u3066 \\(n = p^2 q\\) \u3068\u8868\u3055\u308c\u308b\u3068\u304d, \\(S(n) = 2n\\) \u3092\u6e80\u305f\u3059 \\(n\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(a\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \\(n = 2^a -1\\) \u304c \\(S(n) = n+1\\) \u3092\u6e80\u305f\u3059\u3068\u304d, \\(a\\) \u306f\u7d20\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(a\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b. \\(n = 2^{a-1} \\left( 2^a -1 \\right)\\) \u304c \\(S(n) \\leqq 2n\\) \u3092\u6e80\u305f\u3059\u3068\u304d, \\(n\\) \u306e \\(1\\) \u306e\u4f4d\u306f \\(6\\) \u304b \\(8\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u6761\u4ef6\u3088\u308a\r\n\\[\r\n( 1 +p +p^2 ) ( 1+q ) = 2 p^2 q \\quad ... [1] \\ .\r\n\\]\r\n\\(p\\) \u3068 \\(p^2\\) \u306f\u5947\u5076\u304c\u4e00\u81f4\u3059\u308b\u306e\u3067, \\(1 +p +p^2\\) \u306f\u5947\u6570\u3067\u3042\u308a, \u307e\u305f \\(p^2\\) \u3088\u308a\u5927\u304d\u3044\u306e\u3067, [1] \u304b\u3089\u4ee5\u4e0b\u306e \\(2\\) \u3064\u306e\u5834\u5408\u304c\u8003\u3048\u3089\u308c\u308b.<\/p>\r\n 1*<\/strong>\u3000\\(1 +p +p^2 = q\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(1 +p +p^2 = p^2 q\\) \u306e\u3068\u304d \u3088\u3063\u3066, \u6c42\u3081\u308b \\(n\\) \u306f\r\n\\[\r\nn = 2^2 \\cdot 7 = \\underline{28} \\ .\r\n\\]\r\n (2)<\/strong><\/p>\r\n \\(S(n) = n+1\\) \u3068\u306a\u308b\u306e\u306f, \\(n\\) \u304c\u7d20\u6570\u306e\u3068\u304d\u306a\u306e\u3067, \u300c\\(n = 2^a -1\\) \u304c\u7d20\u6570\u306a\u3089\u3070, \\(a\\) \u306f\u7d20\u6570\u3067\u3042\u308b\u300d... [A] \u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044. 1*<\/strong>\u3000\\(a = 1\\) \u306e\u3068\u304d\r\n\\[\r\nn = 2^1 -1 = 1 \\ .\r\n\\]<\/li>\r\n 2*<\/strong>\u3000\\(a\\) \u304c\u5408\u6210\u6570\u306e\u3068\u304d \u4ee5\u4e0a\u3088\u308a, [A'] \u304c\u793a\u3055\u308c\u3066, [A] \u304c\u6210\u7acb\u3059\u308b\u306e\u3067, \u984c\u610f\u3082\u793a\u3055\u308c\u305f.<\/p>\r\n (3)<\/strong><\/p>\r\n \\(N = 2^a -1\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\nS(n) & \\geqq \\left( 1 +\\cdots +2^{a-1} \\right) ( 1+N ) \\\\\r\n& = \\left( 2^a -1 \\right) 2^a = 2n\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u6761\u4ef6\u3092\u307f\u305f\u3059\u306e\u306f \\(S(n) = 2n\\) \u306e\u3068\u304d\u3067, \u3053\u306e\u3068\u304d, \\(N\\) \u306f\u7d20\u6570\u3067\u3042\u308b. 1*<\/strong>\u3000\\(a = 2\\) \u306e\u3068\u304d\r\n\\[\r\nn = 2^1 ( 2^2 -1 ) = 6 \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(1\\) \u306e\u4f4d\u306f \\(6\\) .<\/p><\/li>\r\n 2*<\/strong>\u3000\\(a\\) \u304c \\(3\\) \u4ee5\u4e0a\u306e\u7d20\u6570\u306e\u3068\u304d \u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(n\\) \u306e\u3059\u3079\u3066\u306e\u6b63\u306e\u7d04\u6570\uff08 \\(1\\) \u3068 \\(n\\) \u3092\u542b\u3080\uff09\u306e\u548c\u3092 \\(S(n)\\) \u3068\u304a\u304f. \u4f8b\u3048\u3070, \\(S(9) = 1 +3 +9 = 13\\) \u3067\u3042\u308b. \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n
\r\n\\[\\begin{align}\r\n1 +q & = 2p^2 \\\\\r\n\\text{\u2234} \\quad q & = 2 p^2 -1 \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, [1] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\n1 +p +p^2 & = 2p^2 -1 \\\\\r\np^2 -p -2 & = 0 \\\\\r\n( p-2 ) ( p+1 ) & = 0 \\\\\r\n\\text{\u2234} \\quad p & = 2 \\quad ( \\ \\text{\u2235} \\ p \\ \\text{\u306f\u7d20\u6570} \\ ) \\ .\r\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\r\nq = 1 +2 +2^2 = 7 \\ .\r\n\\]\r\n\u3053\u308c\u306f, \u7d20\u6570\u3067\u3042\u308b.<\/p><\/li>\r\n
\r\n\\(1 +q = 2\\) \u3067, \\(q\\) \u304c\u7d20\u6570\u3067\u3042\u308b\u3053\u3068\u3068\u77db\u76fe\u3059\u308b\u306e\u3067, \u4e0d\u9069.<\/p><\/li>\r\n<\/ol>\r\n
\r\n[A] \u306e\u5bfe\u5076\u3092\u8003\u3048\u3066, \u300c\\(a\\) \u304c \\(1\\) \u304b\u5408\u6210\u6570\u306a\u3089\u3070, \\(n = 2^a -1\\) \u3082 \\(1\\) \u304b\u5408\u6210\u6570\u3067\u3042\u308b\u300d... [A'] \u3092\u793a\u3059.<\/p>\r\n\r\n
\r\n\\(a = st\\) \uff08\\(s , t\\) \u306f\\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570 ... [2] \uff09\u3068\u8868\u305b\u3066\r\n\\[\\begin{align}\r\nn & = \\left( 2^s \\right)^t -1 \\\\\r\n& = \\underline{\\left( 2^s -1 \\right)} _ {[3]} \\underline{\\left\\{ \\left( 2^s \\right)^{t-1} +\\cdots +1 \\right\\}} _ {[4]} \\ .\r\n\\end{align}\\]\r\n[2] \u3088\u308a, \\([3] \\geqq 2\\) , \\([4] \\geqq 2\\) \u306a\u306e\u3067, \\(n\\) \u306f\u5408\u6210\u6570\u3067\u3042\u308b.<\/p><\/li>\r\n<\/ol>\r\n
\r\n\u3057\u305f\u304c\u3063\u3066 (2)<\/strong> \u306e\u7d50\u679c\u304b\u3089, \\(a\\) \u306f\u7d20\u6570\u3067\u3042\u308b.<\/p>\r\n\r\n
\r\n\\(a\\) \u306f\u5947\u6570\u3067\u3042\u308b.
\r\n\\(2^a\\) \u306e \\(1\\) \u306e\u4f4d\u306f, \\(2 , 4 , 8 , 6\\) \u3092\u5468\u671f \\(4\\) \u3067\u7e70\u308a\u8fd4\u3059\u3053\u3068\u306b\u7740\u76ee\u3059\u308b\u3068, \\(a\\) \u306e \\(4\\) \u3092\u6cd5\u3068\u3059\u308b\u5270\u4f59\u3067\u5206\u985e\u3059\u308b\u3068, \u5404\u6570\u306e \\(1\\) \u306e\u4f4d\u306f, \u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|c|c|c|c} a \\ ( \\mod 4 ) & 2^{a-1} & 2^a & 2^a-1 & n \\\\ \\hline 1 & 6 & 2 & 1 & 6 \\\\ \\hline 3 & 4 & 8 & 7 & 8 \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n\\) \u306e \\(1\\) \u306e\u4f4d\u306f \\(6\\) \u307e\u305f\u306f \\(8\\) .<\/p><\/li>\r\n<\/ol>\r\n