A176613 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.

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, 419, 0, 31, 0, 859, 0, 31, 0, 127, 0, 11, 0, 359, 0, 31, 0, 347, 0, 509, 0, 137, 0, 193, 0, 769, 0, 23, 0, 17

Offset: 0

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 21 2010

Keywords

• nonn

Comments

Let p = prime(i), q = prime(i+1), r = prime(i+2).

(*) p^n + q^n + r^n has to be a prime.

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.

When n is even and p = 3, sequence A160773 gives the even n for which 3^n + 5^n + 7^n is prime.

Examples

			5 + 7 + 11 = 23 = prime(9); 3^2 + 5^2 + 7^2 = 83 = prime(23); 23^3 + 29^3 + 31^3 = 66347 = prime(6616).
		

Links

• Robert Israel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A133530, A133531, A133532, A133533, A160773.

Programs

Maple

f:= proc(n) local p,q,r;
  if n::even then
    if isprime(3^n+5^n+7^n) then return 3
    else return 0
    fi
  fi;
  p:= 2: q:= 3: r:= 5:
  while not isprime(p^n + q^n + r^n) do
    p:= q; q:= r; r:= nextprime(r)
  od;
  p
end proc:
f(0):= 2:
map(f, [$0..100]);

Extensions

a(0) term added by T. D. Noe, Nov 23 2010