
COMMENTS

For n < 200, it is known that a(16) = 16, a(24) = 911, a(32) = 256, a(36) = 911, a(42) = 259, a(48) = 36, a(56) = 11, a(60) = 404, a(66) = 47, a(72) = 29527, a(80) = 83, a(108) = 4, a(110) = 12, a(120) = 12, a(144) = 4, and a(192) = 6631.
Let S = k^n + (k1)^n + ... + 3^n + 2^n. For any n, if k == 1 mod 4 or 2 mod 4, then 2S. Thus, if a(n) is not 0, then a(n) must be congruent to 3 mod 4 or 0 mod 4.
For odd n and for any k > 3, gpf(k1)S, where gpf(x) is the greatest prime factor of x. If k = 3, 3^n + 2^n is always divisible by 5. Thus, a(n) = 0 if n is odd (except a(1) = 2).
For even n < 100, if a(n) was not specified above, a(n) > 100000 or 0.
For even n between 100 and 200 inclusively, if a(n) was not specified above, a(n) > 50000 or 0.


PROG

(PARI) for(k=1, 50000, if(ispseudoprime(sum(i=2, k, i^n)), return(k)))
n=1; while(n<100, print(a(n)); n+=1)
