Problem 489

Erdős Problem #489: declared status 'open'. Formalized: yes. Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let$$B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}.$$If $B=\{b_1<b_2<\cdots\}$ then is it true that$$\lim \frac{1}{x}\sum_{b_i<x}(b_{i+1}-b_i)^2$$exists (and is finite)? Prize: no. Tags: number theory.