Erdős Problem #309

Let \(N\geq 1\). How many integers can be written as the sum of distinct unit fractions with denominators from \(\{1,\ldots,N\}\)? Are there \(o(\log N)\) such integers? If the number of such integers is \(N(n)\) then it is trivial that \(N(n)\leq \log n+O(1)\). Yokota [Yo97] proved that \(N(n)\geq \log n-O(\log\log n)\). Croot [Cr99] proved that every integer at most \[\leq \sum_{n\leq N}\frac{1}{n}-(\tfrac{9}{2}+o(1))\frac{(\log\log N)^2}{\log N}\] can be so represented. If \(F(N)\) counts the number of integers which can be represented in this fashion, then the current best lower bound known is \[F(N) \geq \log N+\gamma -\left(\frac{\pi^2}{3}+o(1)\right)\frac{(\log\log N)^2}{\log N}\] due to Yokota [Yo02]. The answer to the question is no: \(F(N) \sim \log N\). Here \(0\) (the empty sum) is counted.