更新时间:2023-10-21 17:06
亏数是指σ(n)<2n的自然数,其中σ(n)指的是自然数的因数和函数,即自然数n的所有真因数(除去自身)之和。2n-σ(n)称作n的亏度。
例如15的真因数有 1,3,5,1+3+5=9,9<15,所以15是亏数。
最小的一些亏数是: 1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,31,32,33,34,35,37,38,39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,62,63,64,65,67,68,69,71,73,74,75,76,77,79,81,82,83,85,86,87,89,91,92,93,94,95,97,98,99,101,103,105,106,107,109,110,111,113,115,118,119,121,122,123,124,125,127,128,129,131,133,134,135,136,137,139,141,142,143,145,146,147,148,149,151,152,153,154,155,157,158,159,161,163,164,165,166,167,169,170,171,172,173,175,177,178,179,181,183,185,187,188,189,190,191,193,194,195,197,199,201,202,203,205,206,207,209,211,212,213,214,215,217,218,219,221,223,225,226,227,229,230,231,232,233,235,236,237,238,239,241,242,243,244,245,247,248,249,250,251,253,254,255,256,257,259,261,262,263,265,266,267,268,269,271,273,274,275,277,278,279,281,283,284,285,287,289,291,293,295,296,297,298,299,301,302,303,305,307,309,310,311,313,314,315,316,317,319,321,322,323,325,326,327,328,329,331,332,333,334,335,337,338,339,341,343,344,345,346,347,349,351,353,355,356,357,358,359,361,362,363,365,367,369,370,371,373,374,375,376,377,379,381,383,385,386,387,388,389,391,393,394,395,397,399,401,403,405,407,409,410.
奇亏数与偶亏数都有无穷多个,因为每个完全数和亏数的因数(不包括它们自身)都是亏数,所有的质数以及它们的非负整数次幂都是亏数,除6以外的半质数也都是亏数。亏度为1的亏数叫做殆完全数。
与亏数相关的概念是完全数(σ(n) = 2n)和盈数(σ(n)>2n)。最早将自然数分为盈数、完全数和亏数的人是Nicomachus(古希腊数学家尼科马库斯)所著的Introduction Arithmetica(欧洲中世纪数学,于公元前100年写作)。
(1)亏数 例如15的真因数有 1,3,5,1+3+5=9,9<15,所以15是亏数。所以15也是合数当中的亏数。
(2)完备数 例如6的真因数有 1,2,3,1+2+3=6,正好等于6本身,所以6是完备数。所以6也是合数当中的完备数。
(3)盈数 例如12的真因数有 1,2,3,4,6,1+2+3+4+6=16,16>12,所以12是盈数。所以12也是合数当中的盈数。