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

1,1

Comments

Primes p such that both (p-1)/2 and 2*p+1 are prime.

Intersection of A005384 and A005385.

Except for 5, all are congruent to 11 modulo 12.

Primes "inside" Cunningham chains of first kind.

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

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

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

C. K. Caldwell, Cunningham Chains

Example

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

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 *)

Prog

(PARI) forprime(p=2, 1e5, if(isprime(p\2)&&isprime(2*p+1), print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011
(Magma) [p: p in PrimesUpTo(20000) |IsPrime((p-1) div 2) and IsPrime(2*p+1)]; // Vincenzo Librandi, Oct 31 2014
(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

Crossrefs

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330, A156659, A156660, A156877, A162019.

Sequence in context: A097279 A106171 A276174 * A095030 A065114 A309730

Adjacent sequences: A059452 A059453 A059454 * A059456 A059457 A059458

Keyword

nonn

Author

Labos Elemer, Feb 02 2001

Status

approved