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In this note we investigate the set $S(n)$ of positive integer solutions ofthe title Diophantine equation. In particular, for a given $n$ we proveboundedness of the number of solutions, give precise upper bound on the commonvalue of $\sigma_{2}(\overline{X}_{n})$ and $\sigma_{n}(\overline{X}_{n})$together with the biggest value of the variable $x_{n}$ appearing in thesolution. Moreover, we enumerate all solutions for $n\leq 16$ and discuss theset of values of $x_{n}/x_{n-1}$ over elements of $S(n)$.

On the Diophantine equation $\sigma_{2}(\overline{X}_{n})=\sigma_{n}(\overline{X}_{n})$