PROVED
This has been solved in the affirmative.
Let $N\geq 1$. What is the smallest integer not representable as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$? Is it true that the set of integers representable as such has the shape $\{1,\ldots,m\}$ for some $m$?
This was essentially solved by Croot
[Cr99], who proved that if $f(N)$ is the smallest integer not representable then\[\left\lfloor\sum_{n\leq N}\frac{1}{n}-\frac{9}{2}(1+o(1))\frac{(\log\log N)^2}{\log N}\right\rfloor\leq f(N)\]and\[f(N)\leq \left\lfloor\sum_{n\leq N}\frac{1}{n}-\frac{1}{2}(1+o(1))\frac{(\log\log N)^2}{\log N}\right\rfloor.\]It follows that, if $m_N=\lfloor \sum_{n\leq N}\frac{1}{n}\rfloor$, then the set of integers representable is, for all $N$ sufficiently large, either $\{1,\ldots,m_N-1\}$ or $\{1,\ldots,m_N\}$.
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Additional thanks to: Wouter van Doorn
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #308, https://www.erdosproblems.com/308, accessed 2026-04-11
