A219026 Number of primes p<=n such that 2n-p and 2n+p-2 are both prime

0, 0, 1, 0, 2, 0, 1, 2, 1, 0, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 0, 3, 1, 2, 2, 1, 2, 3, 1, 1, 3, 2, 2, 3, 3, 3, 3, 3, 1, 3, 1, 3, 2, 2, 2, 3, 4, 1, 4, 2, 2, 5, 1, 2, 4, 0, 3, 2, 3, 3, 2, 2, 1, 3, 3, 4, 4, 2, 3, 5, 3, 4, 3, 2, 3, 4, 3, 1, 3, 4, 3, 4, 3, 5, 4, 3, 3, 2, 3, 3, 3, 4, 2, 6, 3, 2, 7, 4, 2

Offset

1,5

Comments

Conjecture: a(n)>0 except for n=1,2,4,6,10,22,57.

This is stronger than the Goldbach conjecture; it has been verified for n up to 5*10^7.

Zhi-Wei Sun also conjectured that if n is not among 1,2,3,5,8,87,108 then there is a prime p in (n,2n)

such that 2n-p and 2n+p-2 are both prime. For conjectures in Section 2 of arXiv:1211.1588, he had similar conjectures with p<=n replaced by p in (n,2n)

For example, if n is not among 1,2,4,6,10,15 then there is a prime p in (n,2n) such that

2n-p and 2n+p+2 are both prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000.

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588v2.

Example

a(8)=2 since 3 and 5 are the only primes p<=8 with 16-p and 14+p both prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[2n-Prime[k]]==True&&PrimeQ[2n+Prime[k]-2]==True, 1, 0], {k, 1, PrimePi[n]}]
Do[Print[n, " ", a[n]], {n, 1, 20000}]
np[n_]:=Count[Prime[Range[PrimePi[n]]], _?(AllTrue[{2n-#, 2n+#-2}, PrimeQ]&)]; Array[np, 100] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 23 2017 *)

Crossrefs

Cf. A000040, A002375, A218754, A218585, A218654, A218656, A218825, A219023, A219025.

Sequence in context: A359290 A092928 A321090 * A085097 A374132 A117997

Adjacent sequences: A219023 A219024 A219025 * A219027 A219028 A219029

Keyword

nonn

Author

Zhi-Wei Sun, Nov 10 2012

Status

approved