Erdős [Er81h] notes it is easy to see that there exist $A$ with property $P$, and that any set which increases sufficiently quickly has property $Q$.

He also asks about property $P'$, which is when there are infinitely many $n$ such that $n+a$ is squarefree for all $a\in A$, and property $P'_\infty$, which is when there are infinitely many $n$ such that $n+a$ is squarefree for all but finitely many $a\in A$.

Erdős also asks whether certain special sequences, such as $2^n\pm 1$ or $n!\pm 1$, have properties $P$ or $Q$.

Most of these questions have been resolved by van Doorn and Tao [vDTa25]. In particular they show that any sequence with property $P$ has density $0$, but can have density going to $0$ arbitrarily slowly. They also show that any sequence with property $Q$ has upper density at most $6/\pi^2$, and sequences with property $Q$ exist with density equal to $6/\pi^2$.

They further show that any sequence with properties either $P'$ or $P'_\infty$ have upper density $<6/\pi^2$, and this is best possible in that for any $\epsilon>0$ there exist such sequences with lower density $>6/\pi^2-\epsilon$.

Finally, they also show that $2^n\pm 1$ and $n!\pm 1$ have property $Q$. It remains open whether these sequences have property $P$.

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This page was last edited 02 December 2025.