A298251 The first of three consecutive primes the sum of which is equal to the sum of three consecutive pentagonal numbers.

199, 35951, 46351, 69221, 88427, 230291, 490481, 707573, 829883, 1088419, 1129693, 1258109, 1736101, 1918157, 1976243, 2456939, 2741159, 2753351, 2822881, 3249419, 4603351, 5121713, 5528623, 6186407, 6664429, 6945559, 6964949, 7094839, 7120963, 7147121

Offset: 1

Author

Colin Barker, Jan 15 2018

Keywords

• nonn

Examples

			199 is in the sequence because 199+211+223 (consecutive primes) = 633 = 176+210+247 (consecutive pentagonal numbers).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2352

Crossrefs

Cf. A000040, A000326, A054643, A298073, A298168, A298169, A298222, A298223, A298250.

Programs

Maple

N:= 10^8: # to get all terms where the sums <= N
Res:= NULL:
mmax:= floor((sqrt(8*N-23)-5)/6):
M3:= map(t->9/2*t^2+15/2*t+6, [seq(seq(4*i+j,j=2..3),i=0..mmax/4)]):
for m in M3 do
  r:= ceil((m-8)/3);
  p1:= prevprime(r+1);
  p2:= nextprime(p1);
  p3:= nextprime(p2);
  while p1+p2+p3 > m do
    p3:= p2; p2:= p1; p1:= prevprime(p1);
  od:
  if p1+p2+p3 = m then
    Res:= Res, p1
  fi
od:
Res; # Robert Israel, Jan 16 2018

Mathematica

Module[{nn=50000,pn},pn=Total/@Partition[PolygonalNumber[5,Range[ Ceiling[ (1+Sqrt[1+24 Prime[nn]])/6]]],3,1];Select[Partition[ Prime[ Range[ nn]],3,1],MemberQ[pn,Total[#]]&]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 12 2020 *)

PARI

L=List(); forprime(p=2, 8000000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(72*t-207, &sq) && (sq-15)%18==0, u=(sq-15)\18; listput(L, p))); Vec(L)