A156877 -id:A156877 - cp's OEIS frontend

A059455 Safe primes which are also Sophie Germain primes.

5, 11, 23, 83, 179, 359, 719, 1019, 1439, 2039, 2063, 2459, 2819, 2903, 2963, 3023, 3623, 3779, 3803, 3863, 4919, 5399, 5639, 6899, 6983, 7079, 7643, 7823, 10163, 10799, 10883, 11699, 12203, 12263, 12899, 14159, 14303, 14699, 15803, 17939

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Primes p such that both (p-1)/2 and 2*p+1 are prime.
Except for 5, all are congruent to 11 modulo 12.
Primes "inside" Cunningham chains of first kind.
Infinite under Dickson's conjecture. - Charles R Greathouse IV, Jul 18 2012
See A162019 for the subset of a(n) that are "reproduced" by the application of the transformations (a(n)-1)/2 and 2*a(n)+1 to the set a(n). - Richard R. Forberg, Mar 05 2015

			83 is a term because it is prime and 2*83+1 = 167 and (83-1)/2 = 41 are both primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Chris K. Caldwell, Cunningham Chains.

Intersection of A005384 and A005385.

Cf. A053176, A059452, A059453, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156659, A156660, A156877, A162019.

[p: p in PrimesUpTo(20000) |IsPrime((p-1) div 2) and IsPrime(2*p+1)]; // Vincenzo Librandi, Oct 31 2014

lst={}; Do[p=Prime[n]; If[PrimeQ[(p-1)/2]&&PrimeQ[2*p+1], AppendTo[lst, p]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 02 2008 *)
Select[Prime[Range[1000]], AllTrue[{(# - 1)/2, 2 # + 1}, PrimeQ] &] (* requires Mathematica 10+; Feras Awad, Dec 19 2018 *)

forprime(p=2,1e5,if(isprime(p\2)&&isprime(2*p+1),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

from itertools import count, islice
from sympy import isprime, prime
def A059455_gen(): # generator of terms
    return filter(lambda p:isprime(p>>1) and isprime(p<<1|1),(prime(i) for i in count(1)))
A059455_list = list(islice(A059455_gen(),10)) # Chai Wah Wu, Jul 12 2022

A156660(a(n))*A156659(a(n)) = 1; A156877 gives numbers of these numbers <= n. - Reinhard Zumkeller, Feb 18 2009

A156874 Number of Sophie Germain primes <= n.

Original entry on oeis.org

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

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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A059455 Safe primes which are also Sophie Germain primes.

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A156874 Number of Sophie Germain primes <= n.

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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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