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

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Jan 31 2014

Keywords

• nonn

Comments

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

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

Examples

			a(4) = 1 since 2*4 - 1 = 2 + 2 + 3 with 2 and 3 terms of A234695.
a(5) = 2 since 2*5 - 1 = 2 + 2 + 5 = 3 + 3 + 3 with 2, 3, 5 terms of A234695.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
• H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252 [math.NT], 2012-2013.
• H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013-2014.
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Crossrefs

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

Programs

Mathematica

p[n_]:=PrimeQ[Prime[n]-n+1]
q[n_]:=PrimeQ[n]&&p[n]
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]}]
Table[a[n],{n,1,100}]