Let $\mathbb{N}$ denote the set of positive integers. For every integer $k\in\mathbb{N}$ let $m(k)$ denote the minimal size of a finite set $S\subseteq \mathbb{N}$ such that $\sum\_{j\in S}j^{-1}=k$.


What is the asymptotic growth of $m(k)$?



Size of finite subset of $\mathbb{N}$ such that the sum of reciprocals is a given positive integer
[Ernie Croot](https://www.cambridge.org/core/journals/mathematika/article/on-some-questions-of-erdos-and-graham-about-egyptian-fractions/2A5528F5D4173BCEC7E6FB4469D18812) showed that for all large $N$, every positive integer below 
$$
\sum\_{n\le N} \frac 1n - \Big(\frac{9}{2}+o(1)\Big) \frac{(\log \log N)^2}{\log N}
$$ 
can be represented as a sum of unit fractions with denominator below $N$. Clearly 
no integer larger than $\sum\_{n\le N} 1/n$ can be so represented. This gives the desired asymptotic growth for $m(k)$: namely 
$$ 
m(k) = \exp(k-\gamma+ o(1)). 
$$


