Positive integer solutions of an equation involving Euler, generalized Euler, and Smarandache functions

Abstract This study investigates the positive integer solutions of an equation involving the Smarandache function. The equation is given by tφ ( n ) = φ 2 ( n ) + S ( n 16 ), where φ ( n ) represents the Euler function, φ e ( n ) represents the generalized Euler function with e as a positive integer, and S ( n ) represents the Smarandache function. The solutions of this equation are discussed, and it is proven that the equation only has positive integer solutions when t = 1, 2, 3, 6, 7, 9, 12, 17, 18, 19. Furthermore, all positive integer solutions of the equation are provided.