A156878 -id:A156878 - cp's OEIS frontend

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

Reinhard Zumkeller, Feb 18 2009

nonn

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.

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

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

A156875, A156876, A156877, A156878, A000720.
Cf. A005384 Sophie Germain primes p: 2p+1 is also prime.
Cf. A092816.

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}]]

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

Edited and commented by Daniel Forgues, Jul 31 2009

A059500 Primes p such that both q=(p-1)/2 and 2p + 1 = 4q + 3 are composite numbers. Intersection of A059456 and A053176.

Original entry on oeis.org

13, 17, 19, 31, 37, 43, 61, 67, 71, 73, 79, 97, 101, 103, 109, 127, 137, 139, 149, 151, 157, 163, 181, 193, 197, 199, 211, 223, 229, 241, 257, 269, 271, 277, 283, 307, 311, 313, 317, 331, 337, 349, 353, 367, 373, 379, 389, 397, 401, 409, 421, 433, 439, 449

Offset: 1

Labos Elemer, Feb 05 2001

nonn

Primes which are neither safe nor of Sophie Germain type.
Primes not in Cunningham chains of the first kind. - Alonso del Arte, Jun 30 2005
A010051(a(n))*(1-A156660(a(n)))*(1-A156659(a(n))) = 1; A156878 gives numbers of these numbers <= n. - Reinhard Zumkeller, Feb 18 2009

			Prime p=17 is here because both 35 and 8 are composite numbers. Such primes fall "out of" any Cunningham chain of first kind (or generate Cunningham chains of 0-length).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from _Reinhard Zumkeller)
C. K. Caldwell, Cunningham Chains

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330, A156658.

Complement[Prime[Range[100]], Select[Prime[Range[100]], PrimeQ[2# + 1] &], Select[Prime[Range[100]], PrimeQ[(# - 1)/2] &]] (Delarte)
Select[Prime[Range[100]],!PrimeQ[q=2#+1]&&!PrimeQ[(#-1)/2]&] (* Zak Seidov, Mar 09 2013 *)

is(n)=isprime(n)&&!isprime(n\2)&&!isprime(2*n+1) \\ Charles R Greathouse IV, Jan 16 2013

a(n) ~ n log n. - Charles R Greathouse IV, Jan 16 2013

A156875 Number of safe primes <= n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Reinhard Zumkeller, Feb 18 2009

nonn

a(n) = SUM(A156659(k): 1<=k<=n);

a(n) = A156874(floor((n-1)\2)).

			a(120) = #{5, 7, 11, 23, 47, 59, 83, 107} = 8.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Safe prime

A005385, A156874, A156876, A156877, A156878, A000720.

A156876 Number of primes <= n that are safe primes or Sophie Germain primes.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12

Offset: 1

Reinhard Zumkeller, Feb 18 2009

nonn

			a(120) = #{2,3,5,7,11,23,29,41,47,53,59,83,89,107,113} = 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A156658, A005384, A005385, A000720.

Cf. A156874, A156875, A156877, A156878.

Accumulate[Table[If[AllTrue[{n,2n+1},PrimeQ]||AllTrue[{n,(n-1)/2}, PrimeQ],1,0],{n,100}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 24 2019 *)

a(n) = my(nb=0); forprime(p=2, n, if (isprime(2*p+1) || isprime((p-1)/2), nb++)); nb; \\ Michel Marcus, Nov 06 2022

a(n) = A156874(n)+A156875(n)-A156877(n) = A000720(n)-A156878(n).

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A156874 Number of Sophie Germain primes <= n.

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A059500 Primes p such that both q=(p-1)/2 and 2p + 1 = 4q + 3 are composite numbers. Intersection of A059456 and A053176.

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A156875 Number of safe primes <= n.

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A156876 Number of primes <= n that are safe primes or Sophie Germain primes.

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