An equation involving the Pseudo-Smarandache function and F. Smarandache LCM function

For any positive integer n, the Pseudo-Smarandache function Z(n) is defined as Z(n) = min {m : m [member of] N, n | m (m+1)/2}. And the F. Smarandache function SL(n) is defined as the smallest positive integer k such that n [absolute value of [1, 2, ..., k]], where [1, 2, ..., k] denotes the least common multiple of 1, 2, ..., k. In this paper, we use the elementary methods and congruences to study the solvability of the equation Z(n) + SL(n) = n, and give its all positive integer solutions. Keywords Pseudo-Smarandache function, F. Smarandache LCM function, equation, solution, congruences.
[section] 1. Introduction and results For any positive integer n [greater than or equal to] 1, the Pseudo-Smarandache function Z(n) is defined as the smallest positive integer m such that [...]