Erdős Problem #488 [status

falsifiable; formalized: yes]. Let $A$ be a finite set and$$B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}.$$Is it true that, for every $m>n\geq \max(A)$,$$\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?$$ Current best: The constant $2$ would be the best possible here, as witnessed by taking $A=\{a\}$, $n=2a-1$, and $m=2a$. Prize: no. Tags: number theory.

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