Equal Sum and Product Problem III

Abstract

Denote by $N(n)$ the number of integer solutions $(x_1,x_2,\dots ,x_n)$ of the equation $x_1+x_2+\dots +x_n=x_1{x}_2\cdot \dots \cdot x_n$ such that $x_1\ge x_2\ge \dots \ge x_n\ge 1$, $n\in {\mathbb{Z}}^{+}$. The aim of this paper are is twofold: first we present an asymptotic formula for $\sum _{2\le n\le x}N(n)$, then we verify that the counting function $N(n)$ takes very large value compared to its average value.

The main result of the manuscript has been proved earlier