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

Author

Colin Barker, Jan 19 2018

Keywords

• nonn

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

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