A298463 The first of two consecutive pentagonal numbers the sum of which is equal to the sum of two consecutive primes.

70, 3577, 10795, 36895, 55777, 70525, 78547, 125137, 178365, 208507, 258130, 329707, 349692, 394497, 438751, 468442, 478555, 499105, 619852, 663005, 753667, 827702, 877455, 900550, 1025480, 1085876, 1169092, 1201090, 1211852, 1233520, 1339065, 1508512

Offset

1,1

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Example

70 is in the sequence because 70+92 (consecutive pentagonal numbers) = 162 = 79+83 (consecutive primes).

Mathematica

Select[Partition[PolygonalNumber[5, Range[1500]], 2, 1], CompositeQ[Total[#]/2]&&Total[#] == NextPrime[ Total[#]/2]+NextPrime[Total[#]/2, -1]&][[;; , 1]] (* Harvey P. Dale, Jan 20 2024 *)

Prog

(PARI) L=List(); forprime(p=2, 1600000, q=nextprime(p+1); t=p+q; if(issquare(12*t-8, &sq) && (sq-2)%6==0, u=(sq-2)\6; listput(L, (3*u^2-u)/2))); Vec(L)
(Python)
from __future__ import division
from sympy import prevprime, nextprime
A298463_list, n, m = [], 1 , 6
while len(A298463_list) < 10000:
    k = prevprime(m//2)
    if k + nextprime(k) == m:
        A298463_list.append(n*(3*n-1)//2)
    n += 1
    m += 6*n-1 # Chai Wah Wu, Jan 20 2018

Crossrefs

Cf. A000040, A000326, A001043, A061275, A298462, A298464, A298465, A298466.

Sequence in context: A089274 A266739 A227273 * A275295 A306984 A292986

Adjacent sequences: A298460 A298461 A298462 * A298464 A298465 A298466

Keyword

nonn

Author

Colin Barker, Jan 19 2018

Status

approved