On the solvability of an equation involving the Smarandache function and Euler function

For any positive integer n, let `(n) and S(n) be the Euler function and the Smaran- dache function respectively. In this paper, we use the properties and the curve flgure of these two functions to study the solvability of the equation n P i=1 S(i) = `( n(n+1) 2 ), and prove that this equation has only two positive integer solutions n = 1;10.