I was a kind of curious over this for past few days,
If $a_1, a_2, a_3$ can be any natural nos, what are the distinct integer values $(a_1+a_2+a_3)(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3})$ can attain. Or even if we cannot determine what are these integer values, can we determine wether there are finite or infinitely many of them . My initial thought was also to extend to this to n terms rather than just 3 , but considering n=2 is easily dealt with and for n=3 I am unable to solve, I would like a solution to that first.
Thanks