A298272 The first of three consecutive hexagonal numbers the sum of which is equal to the sum of three consecutive primes.

6, 6216, 7626, 9180, 16836, 19900, 22366, 29646, 76636, 89676, 93096, 114960, 116886, 118828, 322806, 365940, 397386, 422740, 437580, 471906, 499500, 574056, 595686, 626640, 690900, 743590, 984906, 1041846, 1148370, 1209790, 1260078, 1357128, 1450956

Offset: 1

Author

Colin Barker, Jan 16 2018

Keywords

• nonn

Examples

			6 is in the sequence because 6+15+28 (consecutive hexagonal numbers) = 49 = 13+17+19 (consecutive primes).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (first 100 terms from Colin Barker)

Crossrefs

Cf. A000040, A000384, A054643, A298073, A298168, A298169, A298222, A298223, A298250, A298251, A298273.

Programs

Maple

N:= 100: # to get a(1)..a(100)
count:= 0:
mmax:= floor((sqrt(24*N-87)-9)/12):
for i from 1 while count < N do
  mi:= 2*i;
  m:= 6*mi^2+9*mi+7;
  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
    count:= count+1;
    A[count]:= mi*(2*mi-1);
  fi
od:
seq(A[i], i=1..count); # Robert Israel, Jan 16 2018

PARI

L=List(); forprime(p=2, 2000000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(24*t-87, &sq) && (sq-9)%12==0, u=(sq-9)\12; listput(L, u*(2*u-1)))); Vec(L)