cp's OEIS Frontend

A156874 Number of Sophie Germain primes <= n.

%I A156874 #15 May 05 2025 19:10:53
%S A156874 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,
%T A156874 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,
%U A156874 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
%N A156874 Number of Sophie Germain primes <= n.
%C A156874 a(n) = Sum_{k=1..n} A156660(k).
%C A156874 a(n) = A156875(2*n+1).
%C A156874 Hardy-Littlewood conjecture: a(n) ~ 2*C2*n/(log(n))^2, where C2=0.6601618158... is the twin prime constant (see A005597).
%C A156874 The truth of the above conjecture would imply that there exists an infinity of Sophie Germain primes (which is also conjectured).
%C A156874 a(n) ~ 2*C2*n/(log(n))^2 is also conjectured by Hardy-Littlewood for the number of twin primes <= n.
%H A156874 R. Zumkeller, <a href="/A156874/b156874.txt">Table of n, a(n) for n = 1..10000</a>
%H A156874 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SophieGermainPrime.html">Sophie Germain prime</a>
%H A156874 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sophie_Germain_prime">Sophie Germain prime</a>
%F A156874 a(10^n)= A092816(n). - _Enrique Pérez Herrero_, Apr 26 2012
%e A156874 a(120) = #{2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113} = 11.
%t A156874 Accumulate[Table[Boole[PrimeQ[n]&&PrimeQ[2n+1]],{n,1,200}]] (* _Enrique Pérez Herrero_, Apr 26 2012 *)
%t A156874 Accumulate[Table[If[AllTrue[{n,2n+1},PrimeQ],1,0],{n,200}]]
%Y A156874 A156875, A156876, A156877, A156878, A000720.
%Y A156874 Cf. A005384 Sophie Germain primes p: 2p+1 is also prime.
%Y A156874 Cf. A092816.
%K A156874 nonn
%O A156874 1,3
%A A156874 _Reinhard Zumkeller_, Feb 18 2009
%E A156874 Edited and commented by _Daniel Forgues_, Jul 31 2009