On Erdős’ last equation

For a positive integer n ≥ 2 n\ge 2 , we look at the number of solutions of the Diophantine equation n ( x 1 + ⋯ + x n ) = x 1 ⋯ x n . \begin{equation*} n(x_1+\cdots +x_n)=x_1\cdots x_n. \end{equation*} Denoting by f ( n ) f(n) this number we show that lim inf n → ∞ log ⁡ f ( n ) / log ⁡ n = 0 \liminf _{n\to \infty } \log f(n)/\log n=0 .