OPEN
This is open, and cannot be resolved with a finite computation.
Let $f(m)$ be such that if $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert=m$ then every interval in $[1,\infty)$ of length $2N$ contains $\geq f(m)$ many distinct integers $b_1,\ldots,b_r$ where each $b_i$ is divisible by some $a_i\in A$, where $a_1,\ldots,a_r$ are distinct.
Estimate $f(m)$. In particular is it true that $f(m)\ll m^{1/2}$?
Erdős and Sarányi
[ErSa59] proved that $f(m)\gg m^{1/2}$.
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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 #650, https://www.erdosproblems.com/650, accessed 2026-01-16
