一类与伪Smarandache 函数相关的函数方程

首先研究了著名的F.Smarandache函数S(n)的性质,讨论了一类新的包含Smarandache对偶函数及其伪Smarandache函数方程Z(n)+S*(n)-1=kn,k≥1的可解性,利用初等数论及组合方法,结合伪Smarandache函数Z(n)的性质,巧妙地构造了一个新方程。结果给出了这一类方程的所有整数解,即当k=1时,该方程当且仅当有唯一解n=1,当k=2时,仅有解n=2α,α≥1;当k≥3时,无解。从而,本文彻底解决了这类新方程解的问题。

A Function Equation Related to the Pseudo Smarandache Function

First of all,this paper studied the properties of the well-known F.Smarandache function S(n),then studied the positive integer solutions of a new function equation Z(n)+S*(n)-1=kn,k≥1 involving both of the Pseudo Smarandache function and the dual Smarandache function.Used the elementary number theory and combinational method while the property of the Pseudo Smarandache function Z(n),a new equation was made easily.As a result,all positive integer solutions are given for the equation,that was the equation hold if and only if solution n=1 when k=1,and if k=2,it hold solutions n=2α,α≥1,it was no solution if k≥3.So,all the positive integer solutions of this new function equation was solved completely.