  In this note 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 $\sigma_{2}(\overline{X}_{n})$ and $\sigma_{n}(\overline{X}_{n})$
together with the biggest value of the variable $x_{n}$ appearing in the
solution. Moreover, we enumerate all solutions for $n\leq 16$ and discuss the
set of values of $x_{n}/x_{n-1}$ over elements of $S(n)$.
