虧數

在數論中,若一個正整數除了本身之外所有因子之和比此數自身小,則稱此數為虧數(又稱作缺數)。

基本介紹

  • 中文名:虧數
  • 別名:缺數
  • 相關的概念完美數,過剩數
  • 相關單位:虧度
定義,簡介,輕度虧數,

定義

虧數是指使得函式σ(n)< 2n的正整數,其中σ(n)指的是因數和函式,即n的所有正因數(包括n)之和。σ(n)− 2n稱作n虧度

簡介

例如15的真因子有 1,3,5,而1+3+5=9,9<15,所以15可稱為虧數。
最小的一些虧數(OEIS中的數列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, 31,32333435,37,38,39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,62,63,64,65,67,68,69,71,73,74,75,76,77,79,81,82,83,85,86,87,89,91,92,93,94,95,97,98,99,101,103,105,106,107,109,110,111,113,115,118,119,121,122,123,124,125,127,128,129,131,133,134,135,136,137,139,141,142,143,145,146,147,148,149,151,152,153,154,155,157,158,159,161,163,164,165,166,167,169,171,172,173,175,177,178,179,181,183,185,187,188,189,190,191,193,194,195,197,199,201,202,203,205,206,207,209,211,212,213,214,215,217,218,219,221,223,225,226,227,229,230,231,232,233,235,236,237,238,239,241,242,243,244,245,247,248,249,250,251,253,254,255,256,257,259,261,262,263,265,266,267,268,269,271,273,274,275,277,278,279,281,283,284,285,287,289,291,293,295,296,297,298,299,301,302,303,305,307,309,310,311,313,314,315,316,317,319,321,322,323,325,326,327,328,329,331,332,333,334,335,337,338,339,341,343,344,345,346,347,349,351,353,355,356,357,358,359,361,362,363,365,367,369,370,371,373,374,375,376,377,379,381,383,385,386,387,388,389,391,393,394,395,397,399,401,403,405,407,409,410...
奇虧數和偶虧數都有無窮多個,因為顯然所有的素數,以及他們的方冪,都是虧數。另外,每個完美數和虧數的因數(不包括它們自身)都是虧數,所有的半素數也都是虧數。
與虧數相關的概念是完美數σ(n) = 2n)和過剩數σ(n) > 2n)。最早將自然數分為過剩數、完美數和虧數的是Nicomachus所著的Introductio Arithmetica (公元前100年)。
特殊規律:2的n次方的數字的約數當中,除了本身之外,其它約數的和為2的n次方減去1。例如4、8、16等。

輕度虧數

真因子之和比自身小1的數叫做輕度虧數。2的所有次方都是輕度虧數,例如8的因子有1,2,4,而1+2+4=7=8-1,所以8是輕度虧數。

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