PROVED
This has been solved in the affirmative.
Let $A\subseteq\mathbb{N}$ be infinite and $d_A(n)$ count the number of $a\in A$ which divide $n$. Is it true that, for every $k$,\[\limsup_{x\to \infty} \frac{\max_{n<x}d_A(n)}{\left(\sum_{n\in A\cap[1,x)}\frac{1}{n}\right)^k}=\infty?\]
The answer is yes, proved by Erdős and Sárkőzy
[ErSa80].
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This page was last edited 28 October 2025.
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