An equation involving two Smarandache functions and its positive integer solutions

The main purpose is studying the solvability of the equations Z(n)=SL(n) and Z(n)+1=SL(n) involving the Smarandache LCM function SL(n) and the pseudo Smarandache function Z(n).By using the elementary and analytic methods,all positive integer solutions of those equations are obtained.The following two conclusions are proven:(1) For any positive integer n1,the equation Z(n)=SL(n) have positive integer solutions if and only if n=pa·m,where p is odd prime,a≥1 and m1,m|(p~a+1)/2;(2) For any positive integer n1,the equation Z(n)+1=SL(n) have positive integer solutions if and only if n=pa·m,where p is odd prime,a≥1 and m|(p~a-1)/2