In a result I am currently studying (completely unrelated to number theory), I had to examine the solvability of the equation $n = ab+ac+bc$ where $n,a,b,c$ are positive integers $0 < a < b < c.$

As it turned out the set of numbers not expressible in the above way is finite.

Generalizing the equation to four variables and checking the solutions of the equation $n = abc+abd+acd+bcd$ for $0 < a < b < c < d$ I've noticed that it looks like there exists a number $n_0$ such that for $n > n_0$ $n$ is expressible as $abc+abd+acd+bcd.$ The fact that a similar pattern occurs for five variables motivates me to ask the following question:

** Question. ** Given a positive integer $m$ is there a number $n_0$ such that every $n > n_0$ is expressible as $$n = x_1\cdots x_m\left(\frac{1}{x_1} + \cdots + \frac{1}{x_m}\right)$$ where $0 < x_1 < x_2 <\ldots < x_m$.

The question is way too much for my (non-existent) knowledge of number theory. Perhaps there is a known result regarding such equations or, it can be somehow inductively derived from the case $m = 3.$ Any pointers in this direction are appreciated!