In this paper, we investigate the set S (n) of positive integer solutions of the title Diophantine equation. In particular, for a given n we prove boundedness of the number of solutions, give precise upper bound on the common value of σ 2 ( X ¯ n) and σ n ( X ¯ n) together with the biggest value of the variable x n appearing in the solution. Moreover, we enumerate all solutions for n ≤ 1 6 and discuss the set of values of x n / x n − 1 over elements of S (n).