一个包含Smarandache函数的复合函数

A composite function involving the Smarandache function

对任意正整数n,著名的Smarandache函数S(n)定义为最小的正整数m使得n|m!,或者S(n)=min{m∶n|m!,m∈N}.而函数Z(n)定义为最小的正整数k使得n≤k(k+1)/2,即就是Z(n)=min{k∶n≤k(k+1)/2}.本文的主要目的是利用初等及解析方法研究复合函数S(Z(n))的均值,并给出一个较强的渐近公式.

For any positive integer n, the famous Smarandache function S （n） is defined as the smallest positive integer m such that n｜m!, or S（n）=min{m：n｜m!,m∈N}. And the function Z（n） is defined as the smallest positive integer k such that n≤k（k＋1）/2. That is , Z（n）=min{k：n≤k（k＋1）/2}. The main purpose of this paper is using the elementary methods and the analytic methods to study the mean value properties of the compostie function S（Z（n）） ,and give a sharper asymptotic formula for it.