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

0,11

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.

Links

Table of n, a(n) for n=0..79.

Eric Weisstein's World of Mathematics, Twin Primes

Index entries for related partition-counting sequences

Formula

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

Example

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

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]

Prog

(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

Crossrefs

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

Sequence in context: A029249 A025770 A342098 * A099773 A140471 A029061

Adjacent sequences: A283872 A283873 A283874 * A283876 A283877 A283878

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 17 2017

Status

approved