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%I #17 Aug 01 2023 19:30:12
%S 2,5,3,23,0,11,0,5,0,23,3,137,0,5,3,89,0,71,0,17,0,23,0,23,3,131,3,
%T 419,0,31,0,859,0,31,0,127,0,11,0,359,0,31,0,347,0,509,0,137,0,193,0,
%U 769,0,23,0,17
%N Smallest prime p of three consecutive primes such that the sum of their n-th powers is prime, or 0 if such a prime does not exist.
%C Let p = prime(i), q = prime(i+1), r = prime(i+2).
%C (*) p^n + q^n + r^n has to be a prime.
%C When n is even and p > 3, then (*) is composite because primes greater than 3 are either of form 6k-1 or 6k+1 for some k. Hence, squares (or any even power) of such a prime has the form 6k+1. Adding three such even powers will produce a number of the form 6k+3, which is divisible by 3.
%C When n is even and p = 3, sequence A160773 gives the even n for which 3^n + 5^n + 7^n is prime.
%H Robert Israel, <a href="/A176613/b176613.txt">Table of n, a(n) for n = 0..500</a>
%e 5 + 7 + 11 = 23 = prime(9); 3^2 + 5^2 + 7^2 = 83 = prime(23); 23^3 + 29^3 + 31^3 = 66347 = prime(6616).
%p f:= proc(n) local p,q,r;
%p if n::even then
%p if isprime(3^n+5^n+7^n) then return 3
%p else return 0
%p fi
%p fi;
%p p:= 2: q:= 3: r:= 5:
%p while not isprime(p^n + q^n + r^n) do
%p p:= q; q:= r; r:= nextprime(r)
%p od;
%p p
%p end proc:
%p f(0):= 2:
%p map(f, [$0..100]);
%Y Cf. A133530, A133531, A133532, A133533, A160773.
%K nonn
%O 0,1
%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 21 2010
%E a(0) term added by _T. D. Noe_, Nov 23 2010
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