A319250 -id:A319250 - cp's OEIS frontend

A319248 Lesser of the pairs of twin primes in A001122.

3, 11, 59, 179, 347, 419, 659, 827, 1451, 1619, 1667, 2027, 2267, 3467, 3851, 4019, 4091, 4259, 4787, 6779, 6827, 6947, 7547, 8219, 8291, 8819, 9419, 10067, 10091, 10139, 10499, 10859, 12251, 12611, 13931, 14387, 14627, 14867, 16067, 16187, 16979, 17387, 17747

Offset: 1

Jianing Song, Sep 15 2018

nonn

Primes p such that both p and p + 2 are both in A001122.
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.
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 11 is a term.
59 and 61 is a pair of twin primes both having 2 as a primitive root, so 59 is a term.
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.

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

Cf. A001122, A001359, A006512.

A319249 gives p+2, A319250 gives (p-11)/24.

Select[Prime[Range[2^11]], PrimeQ[# + 2] && PrimitiveRoot[#] == 2 && PrimitiveRoot[# + 2] == 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, ", ")))

from itertools import islice
from sympy import isprime, nextprime, is_primitive_root
def A319248_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
A319248_list = list(islice(A319248_gen(),30)) # Chai Wah Wu, Feb 13 2023

a(n) = A319249(n) - 2.

For n >= 2, a(n) = 24*A319250(n-1) + 11.

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.

A377561 Numbers k such that 24k - 1 and 24k + 1 are a pair of twin primes in A115591.

Original entry on oeis.org

8, 13, 62, 78, 113, 125, 132, 157, 207, 230, 315, 337, 428, 473, 493, 570, 652, 763, 788, 902, 928, 932, 987, 1075, 1113, 1135, 1147, 1158, 1225, 1245, 1322, 1327, 1387, 1432, 1483, 1602, 1607, 1672, 1702, 1753, 1767, 1845, 1880, 1973, 1992, 2083, 2155, 2212, 2220, 2233

Offset: 1

Jianing Song, Nov 01 2024

nonn

Numbers k such that 24k - 1 is in A367318. Note that all terms there are congruent to 23 modulo 24.

			8 is a term since the multiplicative order of 2 modulo 24*8 - 1 = 191 is 95, and the multiplicative order of 2 modulo 24*8 + 1 = 193 is 96.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A115591, A002822, A367318, A319250.

isA377561(k) = znorder(Mod(2, 24*k-1))==12*k-1 && znorder(Mod(2, 24*k+1))==12*k \\ No need to check primality as the multiplicative order of 2 modulo a composite odd number m cannot be equal to (m-1)/2; see my comment in A001567

a(n) = (A367318(n) + 1)/24.

cp's OEIS Frontend

A319248 Lesser of the pairs of twin primes in A001122.

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A319249 Greater of the pairs of twin primes in A001122.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A377561 Numbers k such that 24k - 1 and 24k + 1 are a pair of twin primes in A115591.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula