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A319249
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Greater of the pairs of twin primes in A001122.
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3
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5, 13, 61, 181, 349, 421, 661, 829, 1453, 1621, 1669, 2029, 2269, 3469, 3853, 4021, 4093, 4261, 4789, 6781, 6829, 6949, 7549, 8221, 8293, 8821, 9421, 10069, 10093, 10141, 10501, 10861, 12253, 12613, 13933, 14389, 14629, 14869, 16069, 16189, 16981, 17389, 17749
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OFFSET
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1,1
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COMMENTS
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Primes p such that both p - 2 and p are both in A001122.
Apart from the first term, all terms are congruent to 13 mod 24, since terms in A006512 are congruent to 1 mod 6 apart from the first one, and terms in A001122 are congruent to 3 or 5 mod 8.
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.
Also note that a pair of cousin primes can't both appear in A001122, while a pair of sexy primes can.
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LINKS
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FORMULA
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For n >= 2, a(n) = 24*A319250(n-1) + 13.
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EXAMPLE
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11 and 13 is a pair of twin primes both having 2 as a primitive root, so 13 is a term.
59 and 61 is a pair of twin primes both having 2 as a primitive root, so 61 is a term.
Although 137 and 139 is a pair of twin primes, 139 has 2 as a primitive root while 137 doesn't, so 139 is not a term.
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MATHEMATICA
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Select[Prime[Range[2^11]], PrimeQ[# - 2] && PrimitiveRoot[# - 2] == 2 && PrimitiveRoot[#] == 2 &] (* Amiram Eldar, May 02 2023 *)
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PROG
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(PARI) forprime(p=3, 10000, if(znorder(Mod(2, p))==p-1 && znorder(Mod(2, p+2))==p+1, print1(p+2, ", ")))
(Python)
from itertools import islice
from sympy import isprime, nextprime, is_primitive_root
def A319249_gen(): # generator of terms
p = 2
while (p:=nextprime(p)):
if isprime(p+2) and is_primitive_root(2, p) and is_primitive_root(2, p+2):
yield p+2
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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