Let $\ F(n)\ (\mbox{where}\ n\in\mathbb N:=\{1\ 2\ \ldots\})\ $ be the least cardinality $\ |A|\ $ of a set $\ A\subseteq\mathbb N $ such that:


1. $\ \min A=n $
2. $\ \sum\_{x\in A}\frac 1x = 1 $


**QUESTION**   Is set $\ \{n\in\mathbb N:\ \frac{F(n)}n\le 2\}\ $ finite?



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Roughly speaking, $\ \frac {F(n)}n\ \ge\ e-1,\ $ where $\ e:=exp(1)\ $ is one of the Euler's constants, $\ 2.718\ldots.\ $ This induces a bunch of natural questions, I have restricted myself to just one. Please, feel free to enjoy addressing also related questions.



Egyptian representations of $1$
Lots of questions along these lines were raised by Erdos and Graham and many have been solved by Croot, Greg Martin and others. In particular, [Croot](https://eudml.org/doc/279607) has shown that any rational number $r$ can be represented as a 
sum of unit fractions with denominators lying in the interval $[N, (e^r+o(1))N]$ for all large $N$ (indeed his result is more precise, quantifying also the $o(1)$ term). In particular your $F(N)$ is $\sim (e-1)N$ for all large $N$. For related work, see also [Martin](https://arxiv.org/pdf/math/9811112.pdf). 


