This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A283875 #8 Feb 16 2025 08:33:43
%S A283875 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 A283875 23,25,27,31,33,37,40,44,49,52,59,63,69,76,81,90,96,106,114,123,135,
%U A283875 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 A283875 Number of partitions of n into twin primes (A001097).
%C A283875 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 A283875 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%H A283875 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A283875 G.f.: Product_{k>=1} 1/(1 - x^A001097(k)).
%e A283875 a(16) = 4 because we have [13, 3], [11, 5], [7, 3, 3, 3] and [5, 5, 3, 3].
%t A283875 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 A283875 (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 A283875 Cf. A000607, A001097, A077608, A129363, A283876.
%K A283875 nonn
%O A283875 0,11
%A A283875 _Ilya Gutkovskiy_, Mar 17 2017