The number of groups of order $n$ (gnu(n)) can be calculated by a closed formula, if $n$ is squarefree. I could calculate the values with GAP, additionally, I programmed a version in PARI/GP, working fine. I found
and I wonder, whether this is the smallest example $n$ with $gnu(n)>10^6$.
I am looking for a faster way because it takes long (also with GAP) to simply apply brute-force.
It is clear that two prime factors are not enough because $gnu(n)$ is at most $2$ in this case.
If $n$ has $3$ prime factors, of which the smallest is $p$, then $gnu(n)$ is at most $p+4$.
I worked out an upper bound for $4$ prime factors, which I cannot remember, but I am pretty sure, that $4$ factors is not enough either.
Is there an efficient way to calculate the smallest example ?