A084951 Primes in A075893: Primes of the form (p^2+q^2+r^2)/3, where p,q,r are 3 consecutive primes.

113, 193, 577, 1913, 2833, 10753, 44617, 48593, 54617, 69193, 74177, 78593, 86729, 102673, 107873, 122273, 156577, 183497, 214993, 228233, 247697, 308809, 334513, 414313, 581177, 602753, 617369, 636353, 691697, 861857, 1408993, 1786097

Offset: 1

Author

Hugo Pfoertner, Jun 14 2003

Keywords

• easy
• nonn

Comments

With the exception of 2^2+3^2+5^2=38 and 3^2+5^2+7^2=83 all sums of squares of 3 consecutive primes are divisible by 3 because mod(p^2,3)=1 for all primes p>3.

Examples

			a(1)=113 because (7^2+11^2+13^2)/3=(49+121+169)/3=339/3=113 is prime.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A075893, A084952, A133529, A133940.

Programs

Mathematica

b = {}; a = 2; Do[k = (Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a)/3; If[PrimeQ[k], AppendTo[b, n]], {n, 1, 200}]; b (* Artur Jasinski, Sep 30 2007 *)

PARI

v=vector(10000);i=0;p=5;q=7; forprime(r=8,1e8,if(isprime(t=(p^2+q^2+r^2)/3), v[i++]=t; if(i==#v,return)); p=q; q=r) \\ Charles R Greathouse IV, Feb 14 2011

Extensions

Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar.