Related To The Smarandache Function Equation

In this paper, we study the divisibility of the Smarandache function and the solvability of equations related to the Smarandache function. By using the elementary and analytic methods, we establish a divisibility relationship for the Smarandache function. In a similar way, we study the solvability of some equa-tions which are composed with the Smarandache function, Smarandache recip-rocal function, Pseudo-Smarandache function, Pseudo-Smarandachc-square-free function, Dirichlet function, Euler function, and give all the positive integer solutions and the form of these solutions.