cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A307627 Primes p such that 2 is a primitive root modulo p while 8 is not.

Original entry on oeis.org

13, 19, 37, 61, 67, 139, 163, 181, 211, 349, 373, 379, 421, 523, 541, 547, 613, 619, 661, 709, 757, 787, 829, 853, 859, 877, 883, 907, 1117, 1123, 1171, 1213, 1237, 1291, 1381, 1453, 1483, 1531, 1549, 1621, 1669, 1693, 1741, 1747, 1861, 1867, 1987, 2029, 2053
Offset: 1

Views

Author

Jianing Song, Apr 19 2019

Keywords

Comments

Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 3).
According to Artin's conjecture, the number of terms <= N is roughly ((2/5)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N).

Examples

			For p = 67, the multiplicative order of 2 modulo 67 is 66, while 8^22 == 2^(3*22) == 1 (mod 67), so 67 is a term.

Links

Crossrefs

Complement of A019338 with respect to A001122.
Cf. also A005596, A000720, A307628.

Programs

  • Maple
    select(p -> isprime(p) and numtheory:-order(2,p) = p-1,
[seq(i,i=1..10000,6)]); # Robert Israel, Apr 23 2019
  • Mathematica
    Select[Prime@ Range[5, 310], And[FreeQ[#, 8], ! FreeQ[#, 2]] &@ PrimitiveRootList@ # &] (* Michael De Vlieger, Apr 23 2019 *)
  • PARI
    forprime(p=3, 2000, if(znorder(Mod(2, p))==(p-1) && p%3==1, print1(p, ", ")))

A323576 Primes p such that 2 is a primitive root modulo p while 128 is not.

Original entry on oeis.org

29, 197, 211, 379, 421, 491, 547, 659, 701, 757, 827, 883, 1373, 1499, 1667, 1877, 2213, 2269, 2339, 2437, 2549, 2843, 3011, 3067, 3347, 3557, 3571, 3613, 3851, 3907, 4019, 4229, 4243, 4397, 4621, 4691, 4789, 4957, 5573, 5741, 5923, 6203, 6469, 6637, 6763, 6917
Offset: 1

Views

Author

Jianing Song, Aug 30 2019

Keywords

Comments

Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 7).
According to Artin's conjecture, the number of terms <= N is roughly ((6/41)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N).

Links

Crossrefs

Cf. A001122, A005596, A000720.
Primes p such that 2 is a primitive root modulo p and that p == 1 (mod q): A307627 (q=3), A307628 (q=5), this sequence (q=7), A323577 (q=11), A323590 (q=13).

Programs

  • PARI
    forprime(p=3, 7000, if(znorder(Mod(2, p))==(p-1) && p%7==1, print1(p, ", ")))

A323577 Primes p such that 2 is a primitive root modulo p while 2048 is not.

Original entry on oeis.org

67, 419, 661, 859, 947, 1123, 1277, 1453, 2069, 2267, 2333, 2531, 2707, 2861, 3037, 3323, 3499, 3851, 3917, 4093, 4357, 4621, 4973, 5171, 5501, 6029, 6469, 6491, 6733, 7019, 7283, 7349, 7459, 7547, 7789, 7877, 8053, 8669, 8867, 8933, 9901, 9923, 10099, 10253, 10891, 10979
Offset: 1

Views

Author

Jianing Song, Aug 30 2019

Keywords

Comments

Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 11).
According to Artin's conjecture, the number of terms <= N is roughly ((10/109)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N).

Links

Crossrefs

Cf. A001122, A005596, A000720.
Primes p such that 2 is a primitive root modulo p and that p == 1 (mod q): A307627 (q=3), A307628 (q=5), A323576 (q=7), this sequence (q=11), A323590 (q=13).

Programs

  • PARI
    forprime(p=3, 12000, if(znorder(Mod(2, p))==(p-1) && p%11==1, print1(p, ", ")))

A323590 Primes p such that 2 is a primitive root modulo p while 8192 is not.

Original entry on oeis.org

53, 131, 443, 547, 677, 859, 1171, 1301, 1483, 2029, 2237, 2549, 2861, 2939, 3797, 4603, 5227, 5851, 6397, 6709, 6917, 7229, 7307, 7411, 7541, 7853, 8243, 8269, 8867, 8971, 9283, 9491, 9803, 9907, 10037, 10141, 10427, 10973, 11779, 11909, 11987, 12611, 12637, 12923
Offset: 1

Views

Author

Jianing Song, Aug 30 2019

Keywords

Comments

Primes p such that 2 is a primitive root modulo p (i.e., p is in A001122) and that p == 1 (mod 13).
According to Artin's conjecture, the number of terms <= N is roughly ((12/155)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N).

Links

Crossrefs

Cf. A001122, A005596, A000720.
Primes p such that 2 is a primitive root modulo p and that p == 1 (mod q): A307627 (q=3), A307628 (q=5), A323576 (q=7), A323577 (q=11), this sequence (q=13).

Programs

  • Maple
    filter:= proc(p) isprime(p) and numtheory:-order(2,p) = p-1 end proc:
select(filter, [seq(i, i = 1 .. 13000, 26)]); # Robert Israel, Dec 20 2023
  • PARI
    forprime(p=3, 13000, if(znorder(Mod(2, p))==(p-1) && p%13==1, print1(p, ", ")))
Showing 1-4 of 4 results.

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