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A059455 Safe primes which are also Sophie Germain primes. 35
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A097279 A106171 A276174 * A095030 A065114 A309730
KEYWORD
nonn
AUTHOR
Labos Elemer, Feb 02 2001
STATUS
approved

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Last modified June 3 15:36 EDT 2023. Contains 363116 sequences. (Running on oeis4.)