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%I #8 Apr 06 2014 22:15:10
%S 0,0,0,0,1,1,1,1,1,1,1,1,2,2,0,1,2,0,2,3,1,0,2,2,1,3,2,1,1,1,1,2,3,2,
%T 2,0,3,4,2,2,3,2,1,3,4,1,5,2,1,2,3,4,3,1,1,3,2,2,4,3,2,3,3,1,5,5,1,3,
%U 4,2,3,4,4,2,4,2,3,4,2,2
%N Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q, r, pi(p), pi(q), pi(r) all prime, where pi(x) denotes the number of primes not exceeding x (A000720).
%C Conjecture: a(n) > 0 for all n > 36.
%C This is stronger than Goldbach's weak conjecture finally proved by H. A. Helfgott in 2013.
%H Zhi-Wei Sun, <a href="/A237291/b237291.txt">Table of n, a(n) for n = 1..5000</a>
%H H. A. Helfgott, <a href="http://arxiv.org/abs/1205.5252">Minor arcs for Goldbach's problem</a>, arXiv:1205.5252, 2012.
%H H. A. Helfgott, <a href="http://arxiv.org/abs/1305.2897">Major arcs for Goldbach's theorem</a>, arXiv:1305.2897, 2013.
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e a(16) = 1 since 2*16 - 1 = 3 + 11 + 17 with 3, 11, 17, pi(3) = 2, pi(11) = 5 and pi(17) = 7 all prime.
%e a(179) = 1 since 2*179 - 1 = 83 + 83 + 191 with 83, 191, pi(83) = 23 and pi(191) = 43 all prime.
%t p[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
%t a[n_]:=Sum[If[p[2n-1-Prime[Prime[i]]-Prime[Prime[j]]],1,0],{i,1,PrimePi[PrimePi[(2n-1)/3]]},{j,i,PrimePi[PrimePi[(2n-1-Prime[Prime[i]])/2]]}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000040, A000720, A006450, A068307, A230219, A236832, A237284.
%K nonn
%O 1,13
%A _Zhi-Wei Sun_, Feb 06 2014
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