nt.number theory - Bound on the number of solutions of a specific Diophantine equation

Falco had asked a question regarding sum equals to product ( Sum Equals Product)

I have a question in the orthogonal direction. Suppose $X_1,X_2,...,X_n$ are variables and we allow $X_i$'s to take only natural numbers. Look at the following Diophantine equation
$X_1+X_2+ dots + X_n = X_1 X_2 ldots X_n$. Any solution of this equation satiesfies the property that the sum of the entries is equal to their product.

It is easy to see that for every $n$, there are only finitely many solutions of the above equation, denote that number by $f(n)$. It is easy to see that there is no absolute constant $k in mathbb{N}$ such that $f(n) < k$ for every $n$. (look at the sequence $x_n= n!+1$, then $f(x_n) > n$, for $n geq 5$)

If $(x_1,..., x_n)$ is a solution of the above equation then we have $prod_{i=1}^{n-1} x_i < n$. From here one can have a very crude bound for $f(n)$.

Question: 1) What is the best upper bound for $f(n)$?
2) Is there an asymptotic behaviour of $f(n)$ as $n$ tends to infinity.

• Next finite groups - Applications of fusion systems
• Previous gt.geometric topology - Collapsing the medial axis of a polytope