A factoring conversion
The language of tuples $\langle n,i,j,b \rangle$ such that the $j$th bit of the $i$th prime factor of $n$ is $b$ is in NP. In particular, it is in #P. More accurately, there is a #P algorithm that on input $\langle n,i,j \rangle$ gives the $j$th bit of the $i$th prime factor of $n$ (either $0$ or $1$). Since the permanent is #P-complete, we can reduce this algorithm to the computation of some polysize permanent.