OPEN

Let $A\subset\mathbb{N}$ be the set of $n$ such that for every prime $p\mid n$ there exists some $d\mid n$ such that $d\equiv 1\pmod{p}$. Is it true that there exists some constant $c>0$ such that for all large $N$
\[\frac{\lvert A\cap [1,N]\rvert}{N}=\exp(-(c+o(1))\sqrt{\log N}\log\log N).\]

Erdős could prove that there exists some constant $c>0$ such that for all large $N$
\[\exp(-c\sqrt{\log N}\log\log N)\leq \frac{\lvert A\cap [1,N]\rvert}{N}\]
and
\[\frac{\lvert A\cap [1,N]\rvert}{N}\leq \exp(-(1+o(1))\sqrt{\log N\log\log N}).\]
Erdős asked about this because $\lvert A\cap [1,N]\rvert$ provides an upper bound for the number of integers $n\leq N$ for which there is a non-cyclic simple group of order $n$.