Let $N(m)$ be the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $a_i \ (i=1,2,\cdots,n)$ are integers which satisfy $2\le a_1\lt a_2\lt\cdots\lt a_n\le m$.

**Question 1** : What is $N(99)$?

**Question 2** : What is $N(m)$?

**Examples** : I'm going to represent $\sum_{i=1}^n\frac{1}{a_i}$ as $(a_1,a_2,\cdots,a_n).$

The followings are two examples for $(m,n)=(99,42)$.

$(15,17,20,21,22,26,27,30,32,33,34,35,36,38,39,40,42,44,45,48,50,52,54,55,56,60, 63,66,70,75,76,77,78,80,84,85,88,90,91,95,96,99)$

$(17,18,20,21,22,24,26,27,32,33,34,35,36,38,39,40,42,44,45,48,50,52,54,55,56,60,63,66,70,72,75,76,77,78,80,84,85,88,91,95,96,99)$

**Remark** :

Question 1 has been asked on math.SE.

$99$ has no special meaning except that $99$ is not too small and not too large.

Question 2 might be somewhat ambiguous. The best answer would be to represent $N(m)$ by $m$ if it is possible. Also, finding both the max of $m$ and the min of $m$ would be needed.

**Motivation** : The beginning was the following:

"$\sum_{k=2}^n \frac 1k$ is not an integer for any $n$."

(the proof and the other related facts can be seen at https://math.stackexchange.com/questions/494174/proving-that-the-finite-sum-of-the-each-reciprocal-of-any-sequence-of-integers-w).

yourself, you just ignore it. If you close something as "having no research value", it means that the question isbeneathyou and you have a good reason to believe that it is beneath other people here as well (i.e., you can solve it in a few minutes and tell the sketch of a solution in a few lines). I merely do not believe in any other abstract "value" of research or of life, much less in the ability of anybody to evaluate this value at the first glance in non-obvious cases. Voting to reopen :-). $\endgroup$ – fedja Sep 16 '13 at 18:33