A075893 -id:A075893 - 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.

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

Original entry on oeis.org

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

Hugo Pfoertner, Jun 14 2003

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

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

Cf. A075893, A084951.

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

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

A164130 Sums s of squares of three consecutive primes, such that s-+2 are primes.

Original entry on oeis.org

195, 5739, 18459, 32259, 33939, 60291, 74019, 169491, 187131, 244899, 276819, 388179, 783531, 902139, 3588339, 5041491, 5145819, 5193051, 8687091, 9637491, 10227291, 10910019, 11341491, 11757339, 14834379, 15354651, 16115091

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 10 2009

nonn

			5^2 + 7^2 + 11^2 = 195 is a sum of the squared consecutive primes 5, 7 and 11, and 193 and 197 are primes, so 195 is a member of the sequence.

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

Cf. A075893, A133529, A133530, A034962, A164129

q:= 2: r:= 3: R:= NULL: count:= 0:
while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  s:= p^2+q^2+r^2;
  if isprime(s-2) and isprime(s+2) then
    count:= count+1; R:= R,s;
  fi;
od:
R; # Robert Israel, Apr 21 2023

lst={};Do[p=Prime[n]^2+Prime[n+1]^2+Prime[n+2]^2;If[PrimeQ[p-2]&&PrimeQ[p+2], AppendTo[lst,p]],{n,8!}];lst

A133529 INTERSECT A087679. - R. J. Mathar, Aug 27 2009

Comment turned into example by R. J. Mathar, Aug 27 2009

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

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A164130 Sums s of squares of three consecutive primes, such that s-+2 are primes.

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