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A059455 Safe primes which are also Sophie Germain primes.

%I A059455 #69 Mar 15 2025 11:29:50
%S A059455 5,11,23,83,179,359,719,1019,1439,2039,2063,2459,2819,2903,2963,3023,
%T A059455 3623,3779,3803,3863,4919,5399,5639,6899,6983,7079,7643,7823,10163,
%U A059455 10799,10883,11699,12203,12263,12899,14159,14303,14699,15803,17939
%N A059455 Safe primes which are also Sophie Germain primes.
%C A059455 Primes p such that both (p-1)/2 and 2*p+1 are prime.
%C A059455 Except for 5, all are congruent to 11 modulo 12.
%C A059455 Primes "inside" Cunningham chains of first kind.
%C A059455 Infinite under Dickson's conjecture. - _Charles R Greathouse IV_, Jul 18 2012
%C A059455 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
%H A059455 Amiram Eldar, <a href="/A059455/b059455.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A059455 Chris K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=CunninghamChain">Cunningham Chains</a>.
%F A059455 A156660(a(n))*A156659(a(n)) = 1; A156877 gives numbers of these numbers <= n. - _Reinhard Zumkeller_, Feb 18 2009
%e A059455 83 is a term because it is prime and 2*83+1 = 167 and (83-1)/2 = 41 are both primes.
%t A059455 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 *)
%t A059455 Select[Prime[Range[1000]], AllTrue[{(# - 1)/2, 2 # + 1}, PrimeQ] &] (* requires Mathematica 10+; _Feras Awad_, Dec 19 2018 *)
%o A059455 (PARI) forprime(p=2,1e5,if(isprime(p\2)&&isprime(2*p+1),print1(p", "))) \\ _Charles R Greathouse IV_, Jul 15 2011
%o A059455 (Magma) [p: p in PrimesUpTo(20000) |IsPrime((p-1) div 2) and IsPrime(2*p+1)]; // _Vincenzo Librandi_, Oct 31 2014
%o A059455 (Python)
%o A059455 from itertools import count, islice
%o A059455 from sympy import isprime, prime
%o A059455 def A059455_gen(): # generator of terms
%o A059455     return filter(lambda p:isprime(p>>1) and isprime(p<<1|1),(prime(i) for i in count(1)))
%o A059455 A059455_list = list(islice(A059455_gen(),10)) # _Chai Wah Wu_, Jul 12 2022
%Y A059455 Intersection of A005384 and A005385.
%Y A059455 Cf. A053176, A059452, A059453, A059455, A059456, A007700, A005602, A023272, A023302, A023330, A156659, A156660, A156877, A162019.
%K A059455 nonn
%O A059455 1,1
%A A059455 _Labos Elemer_, Feb 02 2001