Does the list of "number of groups of order $n$" contain every natural number?

Question

In other words:

For every natural number $m$, does there always exist an $n$ for which there are exactly $m$ groups of order $n$ up to isomorphism?

Or is this an open question in mathematics? If it is an open question, are there any famous conjectures one way or the other? And what progress has been made in answering the question?

Answer 1

That is not known (as far as I am aware). There is some relevant discussion in the book by Blackburn, Neumann, and Vekataraman "Enumeration of finite groups". The relevant section is $21.6$ "Surjectivity of the enumeration function" on page $238$.

While I have not read through that section, my understanding is that the authors do not provide a definite answer there. (Though they point out that this question has been asked before, several times; see below.) My suspicion is confirmed by the fact that it is repeated as an open problem on page $268$ (Question $22.36$).