Which integers $n>2$ have the following property?


There is a group $G$ such that 


* $G^n \cong G$; and
* for all integers $k$ with $1<k<n$ we have $G^k\not \cong G$.



Groups with $G^n \cong G$ for some integer $n$
This is possible with abelian groups for any $n$; see [this answer](https://mathoverflow.net/a/10194/75) to a very similar question.


