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%I #7 Mar 17 2017 21:56:05
%S 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,
%T 23,25,27,31,33,37,40,44,49,52,59,63,69,76,81,90,96,106,114,123,135,
%U 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
%N Number of partitions of n into twin primes (A001097).
%C 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F G.f.: Product_{k>=1} 1/(1 - x^A001097(k)).
%e a(16) = 4 because we have [13, 3], [11, 5], [7, 3, 3, 3] and [5, 5, 3, 3].
%t 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]
%o (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
%Y Cf. A000607, A001097, A077608, A129363, A283876.
%K nonn
%O 0,11
%A _Ilya Gutkovskiy_, Mar 17 2017
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