Q: Is there a classification of numbers $k$ such that $n^k+k^n$ is not prime for all $n\geq 2$? Say a number $k$ is cool when the number $n^k+k^n$ is not a prime for all $n \geq 2$.

Question 1: Is there a classification of cool numbers?

For example $k=2$ or $k=15$ are not cool but $k=4$ is cool. Do you know some other cool numbers?
Note that $n^4+4^n=(n^2-2^{(n+1)/2}n+2^n)(n^2+2^{(n+1)/2}n+2^n)$ for odd $n$ and for even $n$ the number $n^4+4^n$ is even and thus $k=4$ is cool.

Question 2: Call $k$ eventually cool when $n^k+k^n$ is prime only for finitely many $n \geq 2$. Is there a classification of eventually cool numbers?

Do you know eventually cool numbers that are not cool?

A: Certainly all numbers of the form $4^{2m+1}$ are cool.  When $n = 2h+1$ is odd render
$n^{4^{2m+1}}+(4^{2m+1})^n=a^4+4b^4$
$a=n^{4^{2m}}$
$b=2^{h(2m+1) + m}$
And apply the (Sophie Germain) factorization noted in the question for the specific case $k=4$.

A: If $GCD(n,k) = g \ne 1$, $n^k + k^n$ is never prime because
$$n^k + k^n = g\bigg({n^k + k^n \over g}\bigg)$$
