关于拟亲和数的一个注记

A Note on Quasi-Amicable Number

对于正整数n,如果σ(n)等于2n,则称n为完全数,其中σ(n)为n的所有正约数之和。对于正整数m,n,如果它们各自的所有正约数之和都等于两数之和,则称m和n是一对亲和数。而如果正整数m和n各自的所有正约数之和都等于m+n+1,则称它们为一对拟亲和数。为了判断整数是否为拟亲和数,文章在讨论费玛数和数论函数性质的基础上,找到了一种验证一个整数是否是拟亲和数的方法,从而证明了费玛数不与其他正整数构成拟亲和数对的结论。

For any positive integer n let be the sum of its divisors. A positive integer n such that is called perfect. Two distinct positive integers m and n are called amicable if . And two distinct positive integers m and n are called quasi-amicable if . In this paper, we find out a kind of method of verifying whether an integer is quasi-amicable or not, and prove the Fermat numbers are never perfect or part of an quasi-amicable pair. The discussion is based on some property of the Fermat number and arithmetic function.