A084952 Middle q of three consecutive primes p,q,r such that (p^2 + q^2 + r^2)/3 is prime.

11, 13, 23, 43, 53, 103, 211, 223, 233, 263, 271, 281, 293, 317, 331, 349, 397, 431, 463, 479, 499, 557, 577, 643, 761, 773, 787, 797, 829, 929, 1187, 1327, 1373, 1399, 1427, 1429, 1451, 1453, 1483, 1583, 1627, 1667, 1693, 1747, 1753, 1783, 2027, 2099, 2129

Offset: 1

Author

Hugo Pfoertner, Jun 14 2003

Keywords

• easy
• nonn

Examples

			a(3)=23 because (19^2 + 23^2 + 29^2)/3 = 1731/3 = 577 is prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A075893, A084951.

Programs

Maple

q:= 5: r:= 7:
Res:= NULL: count:= 0:
while count < 100 do
  p:= q;
  q:= r;
  r:= nextprime(r);
  if isprime((p^2+q^2+r^2)/3) then count:= count+1; Res:= Res,q fi
od:
Res; # Robert Israel, Aug 20 2018

Mathematica

Select[Partition[Prime[Range[400]],3,1],PrimeQ[Total[#^2]/3]&][[;;,2]] (* Harvey P. Dale, Sep 08 2023 *)