一类与伪Smarandache函数相关的函数方程

A Function Equation Related to the Pseudo Smarandache Function

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。

First of all, this paper studied the properties of the well-known F. Smarandaehe function S（n） , then studied the positive in- teger 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 Smaran- daehe 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 = 2^α, α≥ 1, it was no solution if k ≥ 3. So, all the positive integer solutions of this new function equation was solved completely.