Erdős Problem #768

declared status 'open'. Formalized: no. Let $A\subset\mathbb{N}$ be the set of $n$ such that for every prime $p\mid n$ there exists some $d\mid n$ with $d>1$ such that $d\equiv 1\pmod{p}$. Is it true that there exists some constant $c>0$ such that for all large $N$$$\frac{\lvert A\cap [1,N]\rvert}{N}=\exp(-(c+o(1))\sqrt{\log N}\log\log N).$$ Current best: Erd\H{o}s 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$$\frac{\lvert A\cap [1,N]\rvert}{N}\leq \exp(-(1+o(1))\sqrt{\log N\log\log N}).$$Erd\H{o}s 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 $n$. Prize: no. OEIS: A001034, A352287. Tags: number theory.

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