TITLE: Asymptotic behaviour of the sequence of the number of groups of order $n$
QUESTION [5 upvotes]: Let $f(n) = \text{# of groups of order} \ n$. I want to study the asymptotic behaviour of this sequence as $n \to \infty$. Clearly $\lim \inf f(n) = 1$ and $\lim \sup f(n) = \infty$, so the sequence is jumping around.
I'm wondering if there exists a "nice" function $g$ such that $$0<\limsup_{n \to +\infty} \frac{f(n)}{g(n)}<+\infty$$

REPLY [3 votes]: It may really depend on the prime factorization of $n$. For example, you can see here that there's a large gap between the number of groups of order $2^{k} - 1$ and $2^{k}$. 
However, there's an asymptotic formula for the number of groups of order $p^{n}$ for a fixed prime $p$:
$$
f(n, p) = p^{(2/27 + o(1))n^{3}}.
$$
See Higman-Sims asymptotic formula.