A156874 Number of Sophie Germain primes <= n.

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

Offset: 1

Author

Reinhard Zumkeller, Feb 18 2009

Keywords

• nonn

Comments

a(n) = Sum_{k=1..n} A156660(k).

a(n) = A156875(2*n+1).

Hardy-Littlewood conjecture: a(n) ~ 2*C2*n/(log(n))^2, where C2=0.6601618158... is the twin prime constant (see A005597).

The truth of the above conjecture would imply that there exists an infinity of Sophie Germain primes (which is also conjectured).

a(n) ~ 2*C2*n/(log(n))^2 is also conjectured by Hardy-Littlewood for the number of twin primes <= n.

Examples

			a(120) = #{2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113} = 11.
		

Links

• R. Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Sophie Germain prime
• Wikipedia, Sophie Germain prime

Crossrefs

A156875, A156876, A156877, A156878, A000720.

Cf. A005384 Sophie Germain primes p: 2p+1 is also prime.

Cf. A092816.

Programs

Mathematica

Accumulate[Table[Boole[PrimeQ[n]&&PrimeQ[2n+1]],{n,1,200}]] (* Enrique Pérez Herrero, Apr 26 2012 *)
Accumulate[Table[If[AllTrue[{n,2n+1},PrimeQ],1,0],{n,200}]]

Formula

a(10^n)= A092816(n). - Enrique Pérez Herrero, Apr 26 2012

Extensions

Edited and commented by Daniel Forgues, Jul 31 2009