Abstract: For any positive integer n, the famous Smarandache power function SP(n) is defined as the smallest positive integer m such that n|mm, where m and n have the same prime divisors. The main purpose of this paper is using the elementary methods to study the positive integer solutions of an equation involving the Smarandache power function SP(n) and obtain some interesting results. At the same time, we give an open problem about the related equation.

Key words: Smarandache power function, equation, positive integer solutions