A239617 Number of ways to write 2*n = p + q with p, q and pi(2*p) - pi(p) all prime, where pi(x) denotes the number of primes not exceeding x.

0, 0, 0, 0, 1, 1, 2, 2, 3, 2, 1, 3, 3, 2, 4, 1, 3, 4, 2, 2, 4, 3, 1, 3, 3, 2, 5, 2, 2, 5, 2, 4, 5, 2, 5, 6, 4, 4, 6, 4, 4, 7, 4, 1, 8, 3, 3, 7, 2, 4, 6, 5, 4, 5, 8, 5, 10, 5, 3, 12, 2, 4, 9, 3, 4, 7, 8, 4, 9, 7, 4, 9, 5, 4, 10, 2, 4, 8, 4, 6

Offset: 1

Author

Zhi-Wei Sun, Mar 22 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 4.

(ii) Each integer n > 5 can be written as p + q (q > 0) with p and pi(2*q) - pi(q) both prime.

(iii) Any integer n > 2 not equal to 11 can be written as p + q with p prime and pi(2*q) - pi(q) a square.

Part (i) is a refinement of Goldbach's conjecture. It implies that there are infinitely many primes p with pi(2*p) - pi(p) prime.

Examples

			a(5) = 1 since 2*5 = 7 + 3 with 7, 3 and pi(2*7) - pi(7) = 6 - 4 = 2 all prime.
a(6) = 1 since 2*6 = 7 + 5 with 7, 5 and pi(2*7) - pi(7) = 2 all prime.
a(11) = 1 since 2*11 = 11 + 11 with 11 and pi(2*11) - pi(11) = 8 - 5 = 3 both prime.
a(16) = 1 since 2*16 = 13 + 19 with 13, 19 and pi(2*13) - pi(13) = 9 - 6 = 3 all prime.
a(23) = 1 since 2*23 = 23 + 23 with 23 and pi(2*23) - pi(23) = 14 - 9 = 5 both prime.
a(44) = 1 since 2*44 = 59 + 29 with 59, 29 and pi(2*59) - pi(59) = 30 - 17 = 13 all prime.
a(166) = 1 since 2*166 = 103 + 229 with 103, 229 and pi(2*103) - pi(103) = 46 - 27 = 19 all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Crossrefs

Cf. A000040, A000720, A002372, A002375, A237284.

Programs

Mathematica

p[n_,k_]:=PrimeQ[PrimePi[2*Prime[k]]-k]&&PrimeQ[2n-Prime[k]]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[2n-1]}]
Table[a[n],{n,1,80}]