A283875 Number of partitions of n into twin primes (A001097).

1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 6, 7, 7, 9, 9, 11, 12, 13, 15, 16, 19, 20, 23, 25, 27, 31, 33, 37, 40, 44, 49, 52, 59, 63, 69, 76, 81, 90, 96, 106, 114, 123, 135, 144, 157, 169, 183, 197, 212, 230, 246, 266, 286, 307, 330, 353, 381, 406, 436, 468, 499, 536, 572, 613, 654, 698, 746, 795, 849, 904, 964

Offset: 0

Author

Ilya Gutkovskiy, Mar 17 2017

Keywords

• nonn

Comments

Conjecture: every number > 7 is the sum of at most 4 twin primes (automatically implies the truth of the first version of the twin prime conjecture). For example: 8 = 5 + 3; 9 = 3 + 3 + 3; 10 = 5 + 5; 11 = 5 + 3 + 3; 12 = 7 + 5, etc.

Examples

			a(16) = 4 because we have [13, 3], [11, 5], [7, 3, 3, 3] and [5, 5, 3, 3].
		

Links

• Eric Weisstein's World of Mathematics, Twin Primes
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000607, A001097, A077608, A129363, A283876.

Programs

Mathematica

nmax = 79; CoefficientList[Series[Product[1/(1 - Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

PARI

Vec(prod(k=1, 79, 1/(1 - (isprime(k) && (isprime(k - 2) || isprime(k + 2)))*x^k)) + O(x^80)) \\ Indranil Ghosh, Mar 17 2017

Formula

G.f.: Product_{k>=1} 1/(1 - x^A001097(k)).