Given a positive integer $n$, let $N(n)$ denote the number of groups of order $n$, up to isomorphism.

Question:Does $N(n)=n$ hold for some $n>1$?

I checked the OEIS-sequence https://oeis.org/A000001 as well as the squarefree numbers in the range $[2,10^6]$ and found no example. Since we have many $n$ with $N(n)<n$ and some $n$ with $N(n) \gg n$, I see no reason why $N(n)=n$ should be impossible for $n>1$.

muchfaster than $2^k$ (something like $2^{2k^3/27}$ if I remember right). Likewise for $N(p^k)$ for any fixed prime $p$ (when $N(p^k)$ grows like $p^{2k^3/27}$). $\endgroup$ – Noam D. Elkies Nov 27 '15 at 2:08