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A319248 Lesser of the pairs of twin primes in A001122.

%I A319248 #39 May 02 2023 02:27:43
%S A319248 3,11,59,179,347,419,659,827,1451,1619,1667,2027,2267,3467,3851,4019,
%T A319248 4091,4259,4787,6779,6827,6947,7547,8219,8291,8819,9419,10067,10091,
%U A319248 10139,10499,10859,12251,12611,13931,14387,14627,14867,16067,16187,16979,17387,17747
%N A319248 Lesser of the pairs of twin primes in A001122.
%C A319248 Primes p such that both p and p + 2 are both in A001122.
%C A319248 Apart from the first term, all terms are congruent to 11 mod 24, since terms in A001359 are congruent to 5 mod 6 apart from the first one, and terms in A001122 are congruent to 3 or 5 mod 8.
%C A319248 Note that "there are infinitely many pairs of twin primes" and "there are infinitely many primes with primitive root 2" are two famous and unsolved problems, so a stronger conjecture implying both of them is that this sequence is infinite.
%C A319248 Also note that a pair of cousin primes can't both appear in A001122, while a pair of sexy primes can.
%H A319248 Amiram Eldar, <a href="/A319248/b319248.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..406 from Jianing Song)
%F A319248 a(n) = A319249(n) - 2.
%F A319248 For n >= 2, a(n) = 24*A319250(n-1) + 11.
%e A319248 11 and 13 is a pair of twin primes both having 2 as a primitive root, so 11 is a term.
%e A319248 59 and 61 is a pair of twin primes both having 2 as a primitive root, so 59 is a term.
%e A319248 Although 101 and 103 is a pair of twin primes, 101 has 2 as a primitive root while 103 doesn't, so 101 is not a term.
%t A319248 Select[Prime[Range[2^11]], PrimeQ[# + 2] && PrimitiveRoot[#] == 2 && PrimitiveRoot[# + 2] == 2 &] (* _Amiram Eldar_, May 02 2023 *)
%o A319248 (PARI) forprime(p=3, 10000, if(znorder(Mod(2,p))==p-1 && znorder(Mod(2,p+2))==p+1, print1(p, ", ")))
%o A319248 (Python)
%o A319248 from itertools import islice
%o A319248 from sympy import isprime, nextprime, is_primitive_root
%o A319248 def A319248_gen(): # generator of terms
%o A319248     p = 2
%o A319248     while (p:=nextprime(p)):
%o A319248         if isprime(p+2) and is_primitive_root(2,p) and is_primitive_root(2,p+2):
%o A319248             yield p
%o A319248 A319248_list = list(islice(A319248_gen(),30)) # _Chai Wah Wu_, Feb 13 2023
%Y A319248 Cf. A001122, A001359, A006512.
%Y A319248 A319249 gives p+2, A319250 gives (p-11)/24.
%K A319248 nonn
%O A319248 1,1
%A A319248 _Jianing Song_, Sep 15 2018