For squarefree $n$, there is a formula allowing to compute the number of groups of order $n$.
I do not think that such a formula exists for cubefree numbers. If a cubefree number $n$ has the property, that there is no prime power $p^k|n$, such that $p^k\equiv 1\ (\ mod\ q\ )$ for some prime $q|n$, then every group of order $n$ is abelian and $n$ is called an abelian number.
In OEIS, I found a slightly different definition of abelian numbers. Is the criterion I mentioned correct ?
The number of abelian groups of order $n$ can be easily calculated (assuming the prime factorization of $n$ is known). But what is the situation for general cubefree numbers $n$ ?
Is the cubefree case easy enough that the number of groups can be efficiently calculated, or are there cubefree numbers $n$ (of course with known factorization), for which the number of groups of order $n$ is unknown ?