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

1,1

Links

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

Example

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

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 *)

Prog

(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)

Crossrefs

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

Sequence in context: A269549 A324436 A209181 * A097732 A305413 A181007

Adjacent sequences: A298248 A298249 A298250 * A298252 A298253 A298254

Keyword

nonn

Author

Colin Barker, Jan 15 2018

Status

approved