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 A230219 #12 Oct 12 2013 02:19:39
%S A230219 0,0,0,0,0,0,1,1,1,2,1,2,5,2,2,4,3,1,4,3,1,4,1,2,5,2,3,4,3,3,8,6,3,12,
%T A230219 6,2,13,3,3,7,6,4,5,4,4,8,7,4,12,7,3,19,6,3,16,5,4,9,5,5,7,10,4,5,8,3,
%U A230219 14,4,3,14,2,5,12,5,2,14,9,2,10,12,4,12,7,6,12,7,9,14,8,6,12,5,4,19,8,4,23,6,3,14
%N A230219 Number of ways to write 2*n + 1 = p + q + r with p <= q such that p, q, r are primes in A230217 and p + q + 9 is also prime.
%C A230219 Conjecture: a(n) > 0 for all n > 6.
%C A230219 This implies Goldbach's weak conjecture for odd numbers and also Goldbach's conjecture for even numbers.
%C A230219 The conjecture also implies that there are infinitely many primes in A230217.
%H A230219 Zhi-Wei Sun, <a href="/A230219/b230219.txt">Table of n, a(n) for n = 1..10000</a>
%H A230219 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.
%e A230219 a(18) = 1 since 2*18 + 1 = 7 + 13 + 17, and 7, 13, 17 are terms of A230217, and 7 + 13 + 9 = 29 is a prime.
%t A230219 RQ[n_]:=PrimeQ[n+6]&&PrimeQ[3n+8]
%t A230219 SQ[n_]:=PrimeQ[n]&&RQ[n]
%t A230219 a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&PrimeQ[Prime[i]+Prime[j]+9]&&SQ[2n+1-Prime[i]-Prime[j]],1,0],{i,1,PrimePi[n-1]},{j,i,PrimePi[2n-2-Prime[i]]}]
%t A230219 Table[a[n],{n,1,100}]
%Y A230219 Cf. A002375, A068307, A230217, A230140, A230141.
%K A230219 nonn
%O A230219 1,10
%A A230219 _Zhi-Wei Sun_, Oct 11 2013