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亏数

在数论中,若一个正整数除了本身之外所有因子之和比此数自身小,则称此数为亏数(又称作缺数)。

亏数是指使得函数σ(n)< 2n的正整数,其中σ(n)指的是因数和函数,即n的所有正因数(包括n)之和。σ(n) 2n称作n亏度

例如15的真因子有 1,3,5,而1+3+5=9,9<15,所以15可称为亏数。

最小的一些亏数(OEIS中的数列A005100)是: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31,32,33,34,35,37,38,39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,62,63,64,65,67,68,69,71,73,74,75,76,77,79,81,82,83,85,86,87,89,91,92,93,94,95,97,98,99,101,103,105,106,107,109,110,111,113,115,118,119,121,122,123,124,125,127,128,129,131,133,134,135,136,137,139,141,142,143,145,146,147,148,149,151,152,153,154,155,157,158,159,161,163,164,165,166,167,169,171,172,173,175,177,178,179,181,183,185,187,188,189,190,191,193,194,195,197,199,201,202,203,205,206,207,209,211,212,213,214,215,217,218,219,221,223,225,226,227,229,230,231,232,233,235,236,237,238,239,241,242,243,244,245,247,248,249,250,251,253,254,255,256,257,259,261,262,263,265,266,267,268,269,271,273,274,275,277,278,279,281,283,284,285,287,289,291,293,295,296,297,298,299,301,302,303,305,307,309,310,311,313,314,315,316,317,319,321,322,323,325,326,327,328,329,331,332,333,334,335,337,338,339,341,343,344,345,346,347,349,351,353,355,356,357,358,359,361,362,363,365,367,369,370,371,373,374,375,376,377,379,381,383,385,386,387,388,389,391,393,394,395,397,399,401,403,405,407,409,410...

奇亏数和偶亏数都有无穷多个,因为显然所有的素数,以及他们的方幂,都是亏数。另外,每个完美数和亏数的因数(不包括它们自身)都是亏数,所有的半素数也都是亏数。

与亏数相关的概念是完美数(σ(n) = 2n)和过剩数(σ(n) > 2n)。最早将自然数分为过剩数、完美数和亏数的是Nicomachus所著的Introductio Arithmetica (公元前100年)。

特殊规律:2的n次方的数字的约数当中,除了本身之外,其它约数的和为2的n次方减去1。例如4、8、16等。

真因子之和比自身小1的数叫做轻度亏数。2的所有次方都是轻度亏数,例如8的因子有1,2,4,而1+2+4=7=8-1,所以8是轻度亏数。


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