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

Author

Labos Elemer, Feb 02 2001

Keywords

• nonn

Comments

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

Examples

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

Links

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

Crossrefs

Intersection of A005384 and A005385.

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

Programs

Magma

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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