关于伪Smarandache函数的一个混合均值

A Hybrid Mean Value of the Pseudo-Smarandache Function

对任意的正整数n,著名的伪Smarandache函数Z(n)定义为最小的正整数m使得n|m(m+1)/2,即Z(n)=min{m:n|m(m+1)/2,m N}.对任意的正整数n,算术函数Ω(n)定义Ω(1)=0,当n>1且n=p1α1·p2α2...pkαk为n的标准分解式时,Ω(n)=α1p1+α2p2+…+αkpk.利用初等方法和解析方法研究了伪Smarandache函数Z(n)与算术函数Ω(n)的混合均值问题,并得到一个较强的渐近公式.

For any positive integer n,the famous Pseudo-Smarandache function Z（ n） is defined as the smallest positive integer m such that n ｜ m（ m ＋ 1） /2,that is Z（ n） = min{ m： n} m（ m ＋ 1） /2,m N}. And for any positive integer n,the arithmetical function Ω（ n） is defined as Ω（ 1） = 0,if n〉1 and n = p1α1·p2α2... pkαkbe the factorization of n into prime powers,Ω（ n） = α1p1＋ α2p2＋ … ＋ αkpk. The elementary method and analytic method were performed to study the hybrid mean value problem involving the Pseudo-Smarandache function Z（ n） and the arithmetical function Ω（ n）,and a shaper asymptotic formula was proposed.