关于F.Smarandache函数与素因数和函数的一个混合均值

Hybrid mean value of F.Smarandache function and the prime factor sum function

对于任意正整数n,若它的标准分解式是n=p1α1p2α2…pαkk,著名的F.Smarandache函数S(n)定义为:存在最小的正整数m,使得n|m!,即:S(n)=min{m:n m!,m∈N},素因数和函数定义为:ω(n)=p1+p2+…+pk,利用初等及解析的方法研究了F.Smarandache函数S(n)与素因数和函数ω(n)的加权均值分布,得到了新混合函数S(n)ω(n)的均值性质,并给出一个有趣的加权均值分布的渐近公式。

For any positive integer n,n=pα11pα22…pαrr decomposes n into prime powers.The F.Smarandache S（n） is defined as the smallest integer m such that nm!.That is S（n）=min{m：nm!,m∈N}.The prime factor sum function is defined as：ω（n）=p1＋p2＋…＋pk,The main purpose of this paper is using the elementary and analytic methods to study the hybrid mean value problem involving the F.Smarandache function S（n）and the prime factor sum function ω（n）.The mean value properties of this new hybrid function S（n）ω（n） are obatained and the paper gives a sharper asymptotic formula for it.