求解两个与Smarandache函数有关的方程

Two Equations 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!.对于任意给定的正整数n,伪Smarandache函数Z(n)定义为最小的正整数m,使得n│1+2+…m=m(m+1)/2.对任意正整数n,伪Smarandache无平方因子函数Zw(n)定义为最小的正整数m,满足n│mn,即Zw(n)=min{m∶m∈N,n│mn}.用初等方法研究了方程S(n)+Z(n)=n和Zw(Z(n))-Z(Zw(n))=0并给出了它们的全部解.

For any positive integer n, the famous Smarandache function S （n） is defined as the smallest positive integer m such that n I m I . The Pseudo-Smarandache function Z（n） is defined to be the smallest positive integer m such that n│1＋2＋…m=m（m＋1）/2. The Pseudo-Smarandache function Zw （n） is defined to be the smallest positive integer m such that n│mn. The main purpose of this paper is to use the elementary method to study the equation S（n）＋Z（n）=n和Zw（Z（n））-Z（Zw（n））=0 and all solutions for them are given.