Abstract For any positive integer n, let S(n) denotes the Smarandache function, and [empty set](n) is the Euler function. The main purpose of this paper is using the elementary method to study the solutions of the equation S(n) = [empty set](n), and give all solutions for it.
Keywords Smrandache function; Equation; Solutions.
[section] 1. Introduction
For any positive integer n, the Smarandache function S(n) is defined as the smallest integer m such that n|m!. From the definition and the properties of S(n), one can easily deduce that if n = [p.sup.[[alpha].sub.1.]sub.1][p.sup.[[alpha].sub.2].sub.2] x x x [p.sup.[[alpha].sub.k].sub.k] is the prime powers factorization of n, then
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About the arithmetical properities of S(n), many people had studied it before, see references ,  and .
If n [greater than or equal to] 1, the Euler function [empty set](n) is defined to be the number of all positive integers not exceeding n, which are relatively prime to n. It is clear that [empty set](n) is a multiplicative function.
In this paper,...