[seqfan] Smallest k > 1 such that n^k+k^n is prime, or 0 if no such k exists

[Felix Fröhlich]

Sequence (with offset 2) starts 3, 2, ?, 24, ?, 54, 69, 2 .....

Question: Is a(4) = 0?

What I found is:
- when k is even, 4^k + k^4 is also even
- when k is odd and not a multiple of 5, the last digit of 4^k is 4 and the
last digit of k^4 is 1, so  4^k + k^4 is a multiple of 5

This leaves the cases where k % 10 = 5. Can such a value for k exist such
that 4^k + k^4 is prime?

Similar question for n = 6: Can an eligible k exist? Or are there proofs
that a(4) = 0 and a(6) = 0? If those questions can be answered, then maybe
this could be added to the OEIS.

Best regards
Felix