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

1,3

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.

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

Formula

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

Example

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

Mathematica

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

Crossrefs

A156875, A156876, A156877, A156878, A000720.

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

Cf. A092816.

Sequence in context: A364800 A157791 A236857 * A294234 A078767 A331137

Adjacent sequences: A156871 A156872 A156873 * A156875 A156876 A156877

Keyword

nonn

Author

Reinhard Zumkeller, Feb 18 2009

Extensions

Edited and commented by Daniel Forgues, Jul 31 2009

Status

approved