Which natural number can be represented as a product of a sum of natural numbers and a sum of their inverses? I. e. does there exist for a natural $n$ a set of natural numbers $\{a_1, a_2,...a_m\}$ such that $n = (a_1 + a_2 + ...+a_m)(\frac{1}{a_1} + \frac{1}{a_2} + ... +\frac{1}{a_m})$? Call $n$ good if such a set exists, they do not have to be distinct.

All $n = k^2$ are good, if n is good then $2n + 2$ is good as well, 10 and 11 are good, 2,3,5,6,7,8 are not. I'm too lazy to check any further especially if there is a solution out there, please point me if you know one.

Does there exist a constant $C$ such that all $n \geq C$ are good? What if I let $a_i$s to be negative?

Now mathoverflow gave me a link to another question Estimating the size of solutions of a diophantine equation which gives 14 as a good number (hard). Moreover there is a link to a very nice paper "An Unusual Cubic Representation Problem" by Andrew Bremner and Allan MacLeod which gives a whole bunch of solutions already for $m = 3$ on page 38 here http://ami.ektf.hu/uploads/papers/finalpdf/AMI_43_from29to41.pdf, for all their $N$ we get $2(N + 3)$ to be good. So if we take $C = 30$, then all even $N \geq C$ are good (needs a clear proof).

**Question**
There is still question about what happens for odd $N$s. And maybe it's worthy to ask if we can restrict $m$. I. e. What is the smallest $k$ such that any good $N$ can be represented as a product above with $k \geq m$?