Summary: Let \(\varphi \left( n \right)\) and \(S\left( n \right)\) be the Euler function and Smarandache function for a positive integer \(n\), respectively. By using elementary methods and techniques, the explicit formula for \(S\left( {{p^\alpha}} \right)\) is obtained, where \(p\) is a prime and \(\alpha \) is a positive integer. As a corollary, some properties for positive integer solutions of the equations \(\varphi \left( n \right) = S\left( {{n^k}} \right)\) or \(\sigma \left( {{2^\alpha}q} \right)/S\left( {{2^\alpha}q} \right)\) are given, where \(q\) is an odd prime and \(\sigma \left( n \right)\) is the sum of different positive factors for \(n\).