Problem 320

Erdős Problem #320: declared status 'open'. Formalized: no. Let $S(N)$ count the number of distinct sums of the form $\sum_{n\in A}\frac{1}{n}$ for $A\subseteq \{1,\ldots,N\}$. Estimate $S(N)$. Current best: Bleicher and Erd\H{o}s [BlEr75] proved the lower bound$$\log S(N)\geq \frac{N}{\log N}\left(\log 2\prod_{i=3}^k\log_iN\right),$$valid for $k\geq 4$ and $\log_kN\geq k$, and also [BlEr76b] proved the upper bound$$\log S(N)\leq \frac{N}{\log N}\left(\log_r N \prod_{i=3}^r \log_iN\right),$$valid for $r\geq 1$ and $\log_{2r}N\geq 1$. (In these bounds $\log_in$ denotes the $i$-fold iterated logarithm.) Bettin, Greni\'{e}, Molteni, and Sanna [BGMS25] improved the lower bound to$$\log S(N) \geq \frac{N}{\log N}\left(2\log 2\left(1-\frac{3/2}{\log_kN}\right)\prod_{i=3}^k\log_iN\right),$$valid for $k\geq 4$ and $\log_kN\geq 3/2$. (In particular this goes to infinity faster than the lower bound of Bleicher and Erd\H{o}s.) See also [321]. Sanna, A lower bound for the number of Egyptian fractions. Prize: no. OEIS: A072207. Tags: number theory, unit fractions.