A298250 The first of three consecutive pentagonal numbers the sum of which is equal to the sum of three consecutive primes.

176, 35497, 45850, 68587, 87725, 229126, 488776, 705551, 827702, 1085876, 1127100, 1255380, 1732900, 1914785, 1972840, 2453122, 2737126, 2749297, 2818776, 3245026, 4598126, 5116190, 5522882, 6180335, 6658120, 6939126, 6958497, 7088327, 7114437, 7140595

Offset

1,1

Links

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

Example

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

Maple

N:= 10^8: # to get all terms where the sums <= N
Res:= NULL:
mmax:= floor((sqrt(8*N-23)-5)/6):
M:= [seq(seq(4*i+j, j=2..3), i=0..mmax/4)]:
M3:= map(m -> 9/2*m^2+15/2*m+6, M):
for i from 1 to nops(M) do
m:= M3[i];
  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, M[i]*(3*M[i]-1)/2;
  fi
od:
Res; # Robert Israel, Jan 16 2018

Mathematica

Module[{prs3=Total/@Partition[Prime[Range[10^6]], 3, 1]}, Select[ Partition[ PolygonalNumber[ 5, Range[ 5000]], 3, 1], MemberQ[ prs3, Total[#]]&]][[All, 1]] (* Harvey P. Dale, Dec 25 2022 *)

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, (3*u^2-u)/2))); Vec(L)

Crossrefs

Cf. A000040, A000326, A054643, A298073, A298168, A298169, A298222, A298223, A298251.

Sequence in context: A229317 A035829 A284073 * A184293 A159426 A009722

Adjacent sequences: A298247 A298248 A298249 * A298251 A298252 A298253

Keyword

nonn

Author

Colin Barker, Jan 15 2018

Status

approved