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

Author

Colin Barker, Jan 15 2018

Keywords

• nonn

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

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)