
COMMENTS

Checking k up through 1024 suggests that the sequence may continue 1, 2, 4, 2, 1, 4, 2, 1, 2, 1, 16, 2, 8, 2, 2, 4, 4, 1, 2, 2, 4, 2, 4, 2, 2, 4, 4, 4, 2, ...
For any a>1 and b>1, a^k + b^k is composite for all odd k>1. Hence if n^k + (n+1)^k is prime then k must be a power of 2.
It is known that a(11) > 2^22. Is it possible that 11^2^m + 12^2^m is composite for all m > 0?


MATHEMATICA

lst={}; For[n=1, n<=100, n++, k=2; While[k<=2^10 && !PrimeQ[n^k+(n+1)^k], k=2*k]; If[k<=2^10, AppendTo[lst, k], AppendTo[lst, 1]]]; lst
