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 $n = x_1\cdots x_m(\frac{1}{x_1} + \cdots + \frac{1}{x_m})$ 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!

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    $\begingroup$ I would write this as asking for a representation with all positive $x_j$ as $$ n = x_1 x_2 \ldots x_m \left( \frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_m} \right) $$ for any $ n > n_0.$ This agrees with your examples for $m=3$ and $m=4.$ Is this what you want? I find your way of writing this and speaking of a set $X$ as clouding the issue. $\endgroup$ – Will Jagy Jul 26 '10 at 18:37
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    $\begingroup$ If we allow equality, the $n_0$ have been conjectured in oeis.org/classic/A027565, which appears to be growing exponentially. $\endgroup$ – tdnoe Jul 26 '10 at 22:32
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    $\begingroup$ By "taken from a set", what you mean is that they're distinct, right? "Taken from a set" doesn't really mean anything. But it probably is not worth worrying about distinctness at first... $\endgroup$ – Harry Altman Jul 11 '11 at 12:51
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    $\begingroup$ This looks like a very difficult problem as the number of variables $m$ is almost the same as the degree $m-1$. It is of similar flavor to the question "Can we write every sufficiently large number as a sum of four cubes?" or "Is $G(k)<100k$ in the Waring problem?" I would be surprised if this problem were resolved in the next 20 years. $\endgroup$ – GH from MO Jul 11 '11 at 20:37
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    $\begingroup$ GH, I must agree, I was just illustrating that the rough density argument may not be enough. I remember, though, R.C. Vaughan telling Kaplansky that the obstruction here could not be detected $p$-adically, and as such this defeated a conjecture in the first edition of his book on the Hardy-Littlewood method. So, in the second edition, on page 127 it says "There are some exceptions to this, see Exercise 5," then Exercise 5 on page 146 is about $x^2 + y^2 + z^9.$ $\endgroup$ – Will Jagy Jul 11 '11 at 22:40

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