On the Diophantine equation ${\sigma }_2({\overline{X}}_n)={\sigma }_n({\overline{X}}_n)$

Abstract

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 $n\le 16$ and discuss the set of values of $x_n/x_{n-1}$ over elements of $S(n)$.

17 pages