How Find the diophantine equation $x_{1}x_{2}x_{3}\cdots x_{n}=x_{1}+x_{2}+\cdots +x_{n}$ 
let $x_{1},x_{2},\cdots,x_{n}$ such 
  $$n\ge 3,x_{1}\le x_{2}\le\cdots x_{n}$$
  $$x_{1}x_{2}x_{3}\cdots x_{n}=x_{1}+x_{2}+\cdots +x_{n}$$
let the number of ordered pairs of postive integers $(x_{1},x_{2},\cdots,x_{n})$ is $f(n)$

we can find the $f(n)$  closed form?  I guess we have  $$f(n)\le n？$$
it is clear 
$$1+2+3=1\cdot 2\cdot 3$$

Observing that, calling $\sigma_i(x_1,\dots,x_n)$ the $i$-th symmetric polynomial then $x_1+\dots+x_n=\sigma_1(x_1,\dots,x_n)$and $x_1x_2\cdots x_n=\sigma_n(x_1,\dots,x_n)$; then looking at the symmetric polynomials properties maybe you can get some informations on what you want.
