A319248 -id:A319248 - cp's OEIS frontend

A319250 Numbers k such that 24k + 11 and 24k + 13 are a pair of twin primes in A001122.

0, 2, 7, 14, 17, 27, 34, 60, 67, 69, 84, 94, 144, 160, 167, 170, 177, 199, 282, 284, 289, 314, 342, 345, 367, 392, 419, 420, 422, 437, 452, 510, 525, 580, 599, 609, 619, 669, 674, 707, 724, 739, 797, 854, 865, 875, 895, 899, 900, 942, 952, 959, 984, 1004, 1080

Offset: 1

Jianing Song, Sep 15 2018

nonn

Numbers k such that 24k + 11 and 24k + 13 are both in A001122. See A319248 and A319249 for detailed information.

			11 and 13 are a pair of twin primes both having 2 as a primitive root, so 0 is a term.
59 and 61 are a pair of twin primes both having 2 as a primitive root, so 2 is a term.
Although 227 and 229 are a pair of twin primes, neither of them has 2 as a primitive root, so 9 is not a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..405 from Jianing Song)

Cf. A001122, A002822, A319248, A319249.

Select[Range[0, 1080], PrimeQ[24*# + 11] && PrimeQ[24*# + 13] && PrimitiveRoot[24*# + 11] == 2 && PrimitiveRoot[24*# + 13] == 2 &] (* Amiram Eldar, May 02 2023 *)

for(k=0, 1000, if(znorder(Mod(2,24*k+11))==24*k+10 && znorder(Mod(2,24*k+13))==24*k+12, print1(k, ", ")))

a(n) = (A319248(n+1) - 11)/24 = (A319249(n+1) - 13)/24.

A319249 Greater of the pairs of twin primes in A001122.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Sep 15 2018

nonn

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.

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..406 from Jianing Song)

Cf. A001122, A001359, A006512.

A319248 gives p-2, A319250 gives (p-13)/24.

Select[Prime[Range[2^11]], PrimeQ[# - 2] && PrimitiveRoot[# - 2] == 2 && PrimitiveRoot[#] == 2 &] (* Amiram Eldar, May 02 2023 *)

forprime(p=3, 10000, if(znorder(Mod(2,p))==p-1 && znorder(Mod(2,p+2))==p+1, print1(p+2, ", ")))

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
A319249_list = list(islice(A319249_gen(),30)) # Chai Wah Wu, Feb 13 2023

a(n) = A319248(n) + 2.

For n >= 2, a(n) = 24*A319250(n-1) + 13.

A367318 Lesser of twin primes p such that p and p+2 are both in A115591.

Original entry on oeis.org

191, 311, 1487, 1871, 2711, 2999, 3167, 3767, 4967, 5519, 7559, 8087, 10271, 11351, 11831, 13679, 15647, 18311, 18911, 21647, 22271, 22367, 23687, 25799, 26711, 27239, 27527, 27791, 29399, 29879, 31727, 31847, 33287, 34367, 35591, 38447, 38567, 40127, 40847, 42071

Offset: 1

Amiram Eldar, Nov 14 2023

nonn

Primes p such that p+2 is also a prime and (p-1)/ord(2, p) = (p+1)/ord(2, p+2) = 2, where ord(2,k) is the multiplicative order of 2 modulo k.
Equivalently, lesser of twin primes p such that ord(2, p+2) = ord(2, p) + 1,
Equal consecutive values in A001917 that correspond to twin primes (p, p+2) are either 1 if p is in A319248, or 2 if p is in this sequence.
Terms are congruent to 23 modulo 24. - Jianing Song, Nov 01 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A001359 and A115591.

Cf. A001122, A001917, A002326, A014664, A319248, A333743.

Select[Prime[Range[2, 4400]], PrimeQ[# + 2] && MultiplicativeOrder[2, # + 2] == MultiplicativeOrder[2, #] + 1 &]

is(n) = isprime(n) && isprime(n+2) && znorder(Mod(2, n + 2)) == znorder(Mod(2, n)) + 1;

cp's OEIS Frontend

A319250 Numbers k such that 24k + 11 and 24k + 13 are a pair of twin primes in A001122.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A319249 Greater of the pairs of twin primes in A001122.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A367318 Lesser of twin primes p such that p and p+2 are both in A115591.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI