一个包含k次补数数列的恒等式

An Identity Involving the k-power Complement Number Sequence

对任意给定的正整数k≥2及任意正整数n,定义n的Smarandachek次补数ak(n)为最小的正整数,使得nak(n)为一个完全k次方幂,即ak(n)=min{u:u.n=mk;u,m∈N},其中N为所有正整数之集合.利用解析方法研究了级数sum from n=1 to +∞ 1/(nak(n))s的敛散性,并给出一个有趣的恒等式.

For any positive integer k≥2 and any positive integer n,we call ak （n） as the Smarandache k-th power complement number of n,if ak（n） is the smallest positive integer such that nak（n） is a perfect k-th power number. That is,ak （n ）=min {u：u . n=mk;u,m ∈ N}. The main purpose of this paper is to study the convergent property of the series ^↑∞∑n 1/（nak（n）^5 using the analytic method, and give an interesting identity.