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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. 9

%I

%S 113,193,577,1913,2833,10753,44617,48593,54617,69193,74177,78593,

%T 86729,102673,107873,122273,156577,183497,214993,228233,247697,308809,

%U 334513,414313,581177,602753,617369,636353,691697,861857,1408993,1786097

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

%C 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 Charles R Greathouse IV, <a href="/A084951/b084951.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 (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 Cf. A075893, A084952, A133529, A133940.

%K easy,nonn

%O 1,1

%A _Hugo Pfoertner_, Jun 14 2003

%E Edited by _N. J. A. Sloane_, Jun 30 2008 at the suggestion of R. J. Mathar.

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Last modified June 18 12:18 EDT 2021. Contains 345102 sequences. (Running on oeis4.)