We study the Diophantine equation \(n^h(x_1 + \ldots + x_n) = x_1 x_2 \cdots x_n,\) where \(h \in \mathbb {Z}_{\ge 0}\) , and \(x_1 \le \ldots \le x_n \in \mathbb {Z}^+\) . We define the solution set ELE(h, n) consisting of all such integer tuples. We derive bounds on \(x_n\) and \(x_{n-1}\) in ELE(h, n). We prove that \(x_1 \cdot \ldots \cdot x_n \le n^h(n^h + 1)(n^h + n - 1)\) . We also show that for \(h = 1\) , \(x_{n-1} \le n + \max _{d \mid 2n^2 - 2n,\, d \le \sqrt{2n^2 - 2n}} d\) , and for \(h \ge 2\) , the inequality \(x_{n-1} \le 2n^h\) holds in ELE(h, n). Moreover, we study the minimal value g(h, n) of the largest coordinate \(x_n\) , and define \(g(h) = \min _{n \ge 2} g(h,n)\) . We prove that \( \liminf _{n \rightarrow \infty } \frac{g(h,n)}{\frac{\log \log n \cdot \log \log \log \log n}{\log \log \log n}} \ge 1. \) We also estimate the growth of the number of solutions. We show that for any \(\varepsilon > 0,\) we have \(\sum _{m \le n} |ELE(h,m)| < n^{h+1+\varepsilon } \) for sufficiently large n, and that \(|ELE(h,n)| > n^{1 - \varepsilon }\) for infinitely many n.
