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

Author

Zhi-Wei Sun, Nov 10 2012

Keywords

• nonn

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.

Examples

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

Crossrefs

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

Programs

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 *)