Number of groups of order n as a series coefficient

Question

Consider the sequence A000001 in oesis.org:

$ g_{n}= $ number of (isomorphism classes of) groups of order n.

Is it known for which $ z $ the generating function $ \sum_{n=1}^{\infty}g_{n}z^{n} $ converges?

Answer 1

It is known (see Known bounds for the number of groups of a given order.) that there is a constant $C$ such that for any $n$, $$g_n\leq n^{C(\log n)^2}.$$ It follows that $\limsup g_n^{1/n}\leq 1.$ On the other hand, clearly $g_n\geq 1$ for all $n\geq 1$, so $\limsup g_n^{1/n}\geq 1$. Thus $\limsup g_n^{1/n}=1$, and so the radius of convergence of $\sum g_nz^n$ is $1$.