Problem 297

Let $N\geq 1$ . How many $A\subseteq \{1,\ldots,N\}$ are there such that $\sum_{n\in A}\frac{1}{n}=1$ ?

Number theory
Unit fractions

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SolvedAnswered

An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Solved, in the site's label, at the level of the exact exponential growth rate. The answer below does not give a finite- $N$ formula or a multiplicative asymptotic.

CFHMPSV24: D. Conlon, J. Fox, X. He, D. Mubayi, H. T. Pham, A. Suk and J. Verstraëte, A question of Erdős and Graham on Egyptian fractions, Discrete Analysis 2025:28. The printed DOI 10.19086/da.154329 resolved on 2026-10-07 to a different article (Magnitude function determines generic finite metric spaces), so the journal's own page is linked.
LiSa24: Y. P. Liu and M. Sawhney, On Further Questions Regarding Unit Fractions, IMRN 2026(2), rnaf382, DOI 10.1093/imrn/rnaf382; available arXiv v1.
St24: S. Steinerberger, On a Problem Involving Unit Fractions, arXiv:2403.17041v5 (2024).

Problem 297

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