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

%I A084952 #11 Sep 08 2023 17:26:50
%S A084952 11,13,23,43,53,103,211,223,233,263,271,281,293,317,331,349,397,431,
%T A084952 463,479,499,557,577,643,761,773,787,797,829,929,1187,1327,1373,1399,
%U A084952 1427,1429,1451,1453,1483,1583,1627,1667,1693,1747,1753,1783,2027,2099,2129
%N A084952 Middle q of three consecutive primes p,q,r such that (p^2 + q^2 + r^2)/3 is prime.
%H A084952 Robert Israel, <a href="/A084952/b084952.txt">Table of n, a(n) for n = 1..10000</a>
%e A084952 a(3)=23 because (19^2 + 23^2 + 29^2)/3 = 1731/3 = 577 is prime.
%p A084952 q:= 5: r:= 7:
%p A084952 Res:= NULL: count:= 0:
%p A084952 while count < 100 do
%p A084952   p:= q;
%p A084952   q:= r;
%p A084952   r:= nextprime(r);
%p A084952   if isprime((p^2+q^2+r^2)/3) then count:= count+1; Res:= Res,q fi
%p A084952 od:
%p A084952 Res; # _Robert Israel_, Aug 20 2018
%t A084952 Select[Partition[Prime[Range[400]],3,1],PrimeQ[Total[#^2]/3]&][[;;,2]] (* _Harvey P. Dale_, Sep 08 2023 *)
%Y A084952 Cf. A075893, A084951.
%K A084952 easy,nonn
%O A084952 1,1
%A A084952 _Hugo Pfoertner_, Jun 14 2003