How many groups of order $n$ with center {e} exist? 
For which numbers $n$ exists a group of order $n$ with center {e} ?
  And how many groups are there for a given order ?

The first such numbers are $6,10,12,14,18,20,21,...$ Groups of order $32,40,64$,
for example have not center {e}. For $n=18$, we have $2$ groups with center {e}
Is there an easy criterion or formula ?
 A: I am pretty sure there is no general criterion or a formula known, especially for finding the number of groups with trivial center.
But there are plenty of examples of families of such numbers $n$. For example $n$ order of any nonabelian simple group, $n = pq$ with $p > q$ primes such that $q \nmid p-1$, etc..
Furthermore, if $G$ and $H$ have trivial center, then so does $G \times H$. Thus the set of such numbers $n$ is closed under multiplication.
Also note that if a group is nilpotent, then it has nontrivial center. There is a simple description of integers $n$ (in terms of prime factorization) such that every group of order $n$ is nilpotent. Hence if there is a group of of order $n$ with trivial center, then $n$ cannot be of this form.
However, there do exist numbers $n$ such that every group of order $n$ has nontrivial center, but not all groups of order $n$ are nilpotent. Some examples are $n = 28, 40, 44, 63, 76, 88, 92, \cdots$ (OEIS) Describing numbers like this does not seem feasible to me.
