A note on the equal sum and product problem with a fixed coefficient

Abstract Let $a\ge 1$ . Denote by $N_a(n)$ the number of solutions of $$ \begin{align*}x_1+x_2+\cdots+x_n=ax_1x_2\cdot\ldots\cdot x_n,\,\, x_1\ge x_2\ge\cdots\ge x_n\ge 1.\end{align*} $$ For fixed $a>1,$ we obtain the asymptotic behavior of $\sum \limits _{a\le n\le x}N_a(n)$ as $x\to \infty $ . We also obtain asymptotics for $\sum \limits _{a\le n\le x}nN_a(n)$ , $\sum \limits _{a\le n\le x}\frac {1}{n}N_a(n)$ , $\sum \limits _{a\le n\le x}\sum \limits _{d\mid n} N_a(d),$ and $\sum \limits _{a\le n\le x}\sum \limits _{d\mid n} N_a(d)\,\tau \!\left (\frac {n}{d}\right )$ . Furthermore, we prove that for every $\varepsilon>0$ and sufficiently large n , the inequality $ N_a(n)<n^{1+\varepsilon } $ holds.