for what $n$, $2^{2^n}-2^{2^{n-1}}+1$ is prime

Question

By checking for first few values, I figured out that the number $$N=2^{2^n}-2^{2^{n-1}}+1$$ is prime for $n=1,2,3$ but for $n=4,5$ it is composite. So are there any specifications on $n$ such that $N$ is prime?