All Random Solved Random Open
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)?
For example, when $A=\{p^2: p\textrm{ prime}\}$ then $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erdős.

See also [208].