Let $q=p^k$ with $p$ a prime number. Let the integer $c=\frac{q^n}{1+n(q^n-1)}$ (it's an integer because of Hamming identity) be a power of $p$ and $c\equiv 1 \pmod{q-1}$.
I have to prove that the integer $c$ is a power of $q$.
Considering the hypothesis I have the fact that $c=p^v$ with $0\le v\le k$. So $p^v \equiv 1 \pmod{q-1}$. I don't have other ideas to continue.
Thanks in advance !
Since $q\geq 2$ we have that $\frac{q^n}{1+n(q^n-1)}$ is less than $1$ for all $n\geq 2$.
So we must only check the case $n=1$ which always yields $1$.