A219026 - OEIS

%I #10 Sep 23 2017 19:58:41

%S 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,

%T 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,

%U 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

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

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

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

%C 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)

%C 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)

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

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

%H Zhi-Wei Sun, <a href="/A219026/b219026.txt">Table of n, a(n) for n = 1..20000</a>.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588v2.

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

%t a[n_]:=a[n]=Sum[If[PrimeQ[2n-Prime[k]]==True&&PrimeQ[2n+Prime[k]-2]==True,1,0],{k,1,PrimePi[n]}]

%t Do[Print[n," ",a[n]],{n,1,20000}]

%t 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 *)

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

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Nov 10 2012