A133940 -id:A133940 - cp's OEIS frontend

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

Hugo Pfoertner, Jun 14 2003

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.

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

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

Cf. A075893, A084952, A133529, A133940.

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 *)

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

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

A075893 Average of three successive primes squared, (prime(n)^2+prime(n+1)^2+prime(n+2)^2)/3, n>=3.

Original entry on oeis.org

65, 113, 193, 273, 393, 577, 777, 1057, 1337, 1633, 1913, 2289, 2833, 3337, 3897, 4417, 4953, 5537, 6153, 7017, 8073, 9177, 10073, 10753, 11313, 12033, 13593, 15353, 17353, 18417, 20097, 21441, 23217, 24673, 26369, 28129, 29953, 31577, 33761

Offset: 3

Zak Seidov, Oct 17 2002

Unlike the average of three successive primes, the average of three successive primes (greater than 3) squared is always integral.

A133529(n)/3, n >= 3. - Artur Jasinski, Sep 30 2007

			a(3)=65 because (prime(3)^2+prime(4)^2+prime(5)^2)/3=(5^2+7^2+11^2)/3=65.

Vincenzo Librandi, Table of n, a(n) for n = 3..5000

Cf. A133529, A084951, A133940.

[(NthPrime(n)^2+NthPrime(n+1)^2+NthPrime(n+2)^2)/3: n in [3..50]]; // Vincenzo Librandi, Aug 21 2018

b = {}; a = 2; Do[k = (Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a)/3; AppendTo[b, k], {n, 3, 50}]; b (* Artur Jasinski, Sep 30 2007 *)
Mean[#]&/@Partition[Prime[Range[3,50]]^2,3,1] (* Harvey P. Dale, Jun 09 2013 *)

a(n) = (prime(n)^2+prime(n+1)^2+prime(n+2)^2)/3, n>=3.

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

cp's OEIS Frontend

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

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