A236832 - OEIS

%I #13 Jun 22 2022 09:29:04

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

%C Conjecture: a(n) > 0 for all n > 3.

%C This is stronger than Goldbach's weak conjecture which was finally proved by H. A. Helfgott in 2013.

%H Zhi-Wei Sun, <a href="/A236832/b236832.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 [math.NT], 2012-2013.

%H 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 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 a(4) = 1 since 2*4 - 1 = 2 + 2 + 3 with 2 and 3 terms of A234695.

%e a(5) = 2 since 2*5 - 1 = 2 + 2 + 5 = 3 + 3 + 3 with 2, 3, 5 terms of A234695.

%t p[n_]:=PrimeQ[Prime[n]-n+1]

%t q[n_]:=PrimeQ[n]&&p[n]

%t 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 Table[a[n],{n,1,100}]

%Y Cf. A000040, A068307, A230219, A234695, A235189.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Jan 31 2014