TITLE: Does there exist a $n \in \mathbb{N},\; n>1$ such that the number of groups of order $n$ equals $n$?
QUESTION [9 upvotes]: Does there exist a $n \in \mathbb{N},\; n>1$ such that the number of groups of order $n$ equals $n$?
Me and my friends were just curious and checked the first 1000 numbers (using the internet).

REPLY [13 votes]: We do not know.
Let $a(n)$ be number of groups of order $n$. There is a conjecture from Conway, Dietrich and O'Brien (see also O'Brien's website here) that the sequence $$n, a(n), a(a(n)), a(a(a(n))), \dots$$
eventually consists of an infinite sequence of $1$. 
The number $n$ you are looking for would be a counterexample to this conjecture, which is currently open (and fully checked up to $2047$).