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