Let it be $S=\{O_1,O_2,...O_n\}$ a set of an odd number of distinct odd integers, and $O_j\notin S$ another odd integer.
I want to prove (or disprove) that it does not exist any solution such that $$\frac{O_j-2}{O_j} = \sum_{k=1}^n{\frac{1}{O_k}}$$
Any idea?
I guess that the underlying reason for the possible inexistence of solution may be that $O_j-2$ and $O_j$ are consecutive odd integers, and that $\gcd(O_j-2,O_j)=1$, but I can not imagine a way to prove it or disprove it, and I do not find any counterexample.
Thanks in advance!
$${1\over3}={1\over5}+{1\over9}+{1\over45}$$
Since someone downvoted this, maybe I have to explain.
$O_1=5$ is an odd integer.
$O_2=9$ is another odd integer.
$O_3=45$ is yet another odd integer.
$3$ is still another odd integer.
$${1\over5}+{1\over9}+{1\over45}={9\over45}+{5\over45}+{1\over45}={15\over45}={1\over3}={3-2\over3}$$
This establishes that the equation OP is asking about does, in fact, have a solution.
Capiche?
[Thanks to Jose for pointing out a typo, now corrected.]