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

abstract-algebra group-theory finite-groups open-problem groups-enumeration

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 $22.36$).

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