I have a question that I've been thinking for a while now.


*Can you find a set of distinct positive odd integers* $n\_1, n\_2, \ldots, n\_k$ *for some finite positive integer* $k$ *such that* $\left(\frac{1}{n\_1} + \frac{1}{n\_2} + \ldots + \frac{1}{n\_k} \right)$ *is a positive integer as well?* 


This statement obviously holds if we allow $n\_1 \ldots n\_k$ to be even. I'd be glad if you can recommend some articles that study this particular problem.



The sum of reciprocals of odd numbers
In fact, you can choose the $n\_i$ so that the sum is $1$: see the discussion at <http://www.ics.uci.edu/~eppstein/numth/egypt/odd-one.html>


