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

%I A084951 #13 Aug 28 2020 03:06:34
%S A084951 113,193,577,1913,2833,10753,44617,48593,54617,69193,74177,78593,
%T A084951 86729,102673,107873,122273,156577,183497,214993,228233,247697,308809,
%U A084951 334513,414313,581177,602753,617369,636353,691697,861857,1408993,1786097
%N A084951 Primes in A075893: Primes of the form (p^2+q^2+r^2)/3, where p,q,r are 3 consecutive primes.
%C A084951 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.
%H A084951 Charles R Greathouse IV, <a href="/A084951/b084951.txt">Table of n, a(n) for n = 1..10000</a>
%e A084951 a(1)=113 because (7^2+11^2+13^2)/3=(49+121+169)/3=339/3=113 is prime.
%t A084951 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 *)
%o A084951 (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
%Y A084951 Cf. A075893, A084952, A133529, A133940.
%K A084951 easy,nonn
%O A084951 1,1
%A A084951 _Hugo Pfoertner_, Jun 14 2003
%E A084951 Edited by _N. J. A. Sloane_, Jun 30 2008 at the suggestion of R. J. Mathar.