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Numbers n such that n^k + (n-1)^k + ... + 3^k + 2^k is prime for some natural number k.
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%I #29 May 17 2014 02:57:26

%S 2,3,4,7,8,11,12,16

%N Numbers n such that n^k + (n-1)^k + ... + 3^k + 2^k is prime for some natural number k.

%C a(9) > 19. See A240766 for more information.

%C a(n) is also the n-values such that A240766(n) is nonzero.

%C It is known that a(n) must be == 3 mod 4 or 0 mod 4 (except a(1) = 2) due to the parity of the sum. If an n-value is congruent to 1 mod 4 or 2 mod 4, the sum will always be even and thus, not prime.

%C It is known that 31, 36, 40, 43, 47, 56, 67, 83, and 171 are members of this sequence.

%C If n-1 is not squarefree, then n is not a member of this sequence.

%e 2^k is prime for at least one k (and only one k in this instance; k = 1). Thus, 2 is a member of this sequence.

%e 3^k+2^k is prime for at least one k (see A082101). Thus, 3 is a member of this sequence.

%o (PARI) a(n)=for(k=1,4000,if(ispseudoprime(sum(i=2,n,i^k)),return(k)))

%o n=1; while(n<200,if(a(n),print(a(n)));n+=1)

%Y Cf. A081507, A082101, A240766.

%K nonn,hard,more

%O 1,1

%A _Derek Orr_, Apr 12 2014