Is $n=8{,}574{,}796{,}230$ the smallest squarefree number $n$ with $gnu(n)>10^6$?

Question

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

$$gnu(8{,}574{,}796{,}230)=1{,}243{,}776$$

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 ?

Answer 1

Yes, $n=8574796230$ is the smallest square-free number $n$ such that the number of groups of order $n$ is greater than $10^6$.

The number of groups of this order can be calculated in GAP as follows:

gap> NrSmallGroups(8574796230);
1243776

I did not have a chance to try anything better than a brute force approach, or even parallelising the check, so I was patient enough to run the following script to check that there are no smaller numbers with this property:

IsWorthTrying:=function(n) 
local d,x; 
d :=Factors(n); 
if Length(d)>4 and ForAll(Collected(d),x -> x[2]=1) then 
  return true; 
else 
  return false; 
fi;
end;

k:=1;
repeat 
  k:=k+1; 
  if IsWorthTrying(k) then 
    nr:=NrSmallGroups(k); 
    Print(k, " : ", nr, "                 \r");  
  else 
    nr:=0; 
  fi; 
until nr > 10^6; 
Print("\n\n");

As @Peter said, "the smallest solution must have at least $5$ prime factors", so it was checking only those with 5 or more prime factors. It was running from December 16th till February 6th, except maybe several days when it was stopped and then restarted from the point where it was at the time of the shutdown.