A237291 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).

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, 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, 4, 2, 3, 4, 4, 2, 4, 2, 3, 4, 2, 2

Offset: 1

Author

Zhi-Wei Sun, Feb 06 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 36.

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

Examples

			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.
a(179) = 1 since 2*179 - 1 = 83 + 83 + 191 with 83, 191, pi(83) = 23 and pi(191) = 43 all prime.
		

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, 2012.
• H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897, 2013.
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A000720, A006450, A068307, A230219, A236832, A237284.

Programs

Mathematica

p[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
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]]}]
Table[a[n],{n,1,80}]