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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?

Background:

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.

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  • $\begingroup$ @FedorPetrov, thank you for fixing my typo (sloppy me). $\endgroup$ – Wlod AA May 23 '18 at 7:04
  • $\begingroup$ Have you calculated a few values of $F(n)$ and then consulted the Online Encyclopedia of Integer Sequences? $\endgroup$ – Gerry Myerson May 23 '18 at 13:10
  • $\begingroup$ @GerryMyerson, I was hoping that some numerically literate people will provide an initial and relatively extensive table of F(n). Myself, I did something old fashion in a precomputer style like a reincarnation of a 17th century Basho's frog. You decide how promising is my (very fractional) answer below--I hope that it will get amplified in capable hands, and it can be followed by more numerical activities. $\endgroup$ – Wlod AA May 24 '18 at 2:28
  • $\begingroup$ All you need is the first few terms, then you look it up at the OEIS. Exceptional levels of numerical literacy not required. $\endgroup$ – Gerry Myerson May 24 '18 at 2:59
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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 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.

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  • $\begingroup$ Lucia, super! Do these authors mentioned by you illustrate their results on concrete numerical examples? (possibly, they are too advanced for this :) ) --- Myself, I'll make a certain connection with other equally ancient topics. $\endgroup$ – Wlod AA May 24 '18 at 2:27

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