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

1,7

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.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A162915 A359791 A242266 * A304737 A374516 A369028

Adjacent sequences: A239614 A239615 A239616 * A239618 A239619 A239620

Keyword

nonn

Author

Zhi-Wei Sun, Mar 22 2014

Status

approved