

A283875


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


4



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
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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 partitioncounting 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: A265410 A029249 A025770 * A099773 A140471 A029061
Adjacent sequences: A283872 A283873 A283874 * A283876 A283877 A283878


KEYWORD

nonn


AUTHOR

Ilya Gutkovskiy, Mar 17 2017


STATUS

approved



