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A236832 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q and r terms of A234695.

%I A236832 #13 Jun 22 2022 09:29:04
%S A236832 0,0,0,1,2,2,2,2,3,2,3,3,2,4,3,3,4,3,4,4,4,4,3,5,5,7,6,3,5,4,5,4,5,6,
%T A236832 6,6,3,5,7,6,6,3,5,8,8,8,6,7,8,7,6,5,8,9,10,5,7,9,10,11,5,8,9,9,11,6,
%U A236832 8,9,10,8,2,9,10,9,11,6,8,11,12,7,7,10,9,10,8,7,11,10,11,6,8,12,14,13,8,10,11,12,12,10
%N A236832 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q and r terms of A234695.
%C A236832 Conjecture: a(n) > 0 for all n > 3.
%C A236832 This is stronger than Goldbach's weak conjecture which was finally proved by H. A. Helfgott in 2013.
%H A236832 Zhi-Wei Sun, <a href="/A236832/b236832.txt">Table of n, a(n) for n = 1..5000</a>
%H A236832 H. A. Helfgott, <a href="http://arxiv.org/abs/1205.5252">Minor arcs for Goldbach's problem</a>, arXiv:1205.5252 [math.NT], 2012-2013.
%H A236832 H. A. Helfgott, <a href="http://arxiv.org/abs/1305.2897">Major arcs for Goldbach's theorem</a>, arXiv:1305.2897 [math.NT], 2013-2014.
%H A236832 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014-2016.
%e A236832 a(4) = 1 since 2*4 - 1 = 2 + 2 + 3 with 2 and 3 terms of A234695.
%e A236832 a(5) = 2 since 2*5 - 1 = 2 + 2 + 5 = 3 + 3 + 3 with 2, 3, 5 terms of A234695.
%t A236832 p[n_]:=PrimeQ[Prime[n]-n+1]
%t A236832 q[n_]:=PrimeQ[n]&&p[n]
%t A236832 a[n_]:=Sum[If[p[Prime[i]]&&p[Prime[j]]&&q[2n-1-Prime[i]-Prime[j]],1,0],{i,1,PrimePi[(2n-1)/3]},{j,i,PrimePi[(2n-1-Prime[i])/2]}]
%t A236832 Table[a[n],{n,1,100}]
%Y A236832 Cf. A000040, A068307, A230219, A234695, A235189.
%K A236832 nonn
%O A236832 1,5
%A A236832 _Zhi-Wei Sun_, Jan 31 2014