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 1-3 of 3 results.

A323617 Primes p such that 3 is a primitive root modulo p while 243 is not.

Original entry on oeis.org

31, 101, 211, 281, 331, 401, 461, 521, 571, 631, 641, 691, 701, 751, 811, 821, 881, 941, 1061, 1231, 1291, 1301, 1361, 1481, 1601, 1721, 1831, 1901, 1951, 2011, 2081, 2141, 2311, 2371, 2381, 2731, 2741, 2801, 2861, 3041, 3271, 3331, 3391, 3461, 3571, 3581, 3701, 3761, 3821, 3931
Offset: 1

Views

Author

Jianing Song, Aug 30 2019

Keywords

Comments

Primes p such that 3 is a primitive root modulo p (i.e., p is in A019334) and that p == 1 (mod 5).
According to Artin's conjecture, the number of terms <= N is roughly ((4/19)*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. A019334, A005596, A000720.
Primes p such that 3 is a primitive root modulo p and that p == 1 (mod q): A323594 (q=3), this sequence (q=5), A323628 (q=7).

Programs

  • PARI
    forprime(p=5, 4000, if(znorder(Mod(3, p))==(p-1) && p%5==1, print1(p, ", ")))

A323628 Primes p such that 3 is a primitive root modulo p while 2187 is not.

Original entry on oeis.org

29, 43, 113, 127, 197, 211, 281, 379, 449, 463, 617, 631, 701, 953, 1373, 1709, 1723, 2129, 2143, 2213, 2311, 2381, 2549, 2633, 2647, 2731, 2801, 2969, 3137, 3389, 3557, 3571, 3823, 4159, 4229, 4243, 4327, 4397, 4481, 4649, 4663, 4817, 4831, 4999, 5237, 5419
Offset: 1

Views

Author

Jianing Song, Aug 30 2019

Keywords

Comments

Primes p such that 3 is a primitive root modulo p (i.e., p is in A019334) 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. A019334, A005596, A000720.
Primes p such that 3 is a primitive root modulo p and that p == 1 (mod q): A323594 (q=3), A323617 (q=5), this sequence (q=7).

Programs

  • Maple
    select(p -> isprime(p) and numtheory:-order(3,p)=p-1, [seq(i,i=1..10000,7)]); # Robert Israel, Sep 01 2019
  • PARI
    forprime(p=5, 5500, if(znorder(Mod(3, p))==(p-1) && p%7==1, print1(p, ", ")))

A019353 Primes with primitive root 27.

Original entry on oeis.org

2, 5, 17, 29, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269, 281, 293, 317, 353, 389, 401, 449, 461, 509, 521, 557, 569, 593, 617, 641, 653, 677, 701, 773, 797, 809, 821, 857, 881, 929, 941, 953, 977, 1013, 1049, 1061, 1097, 1109, 1193, 1217, 1229, 1277, 1301
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

From Jianing Song, May 12 2024: (Start)
Members of A019334 that are not congruent to 1 mod 3. Terms greater than 2 are congruent to 5 modulo 12.
According to Artin's conjecture, the number of terms <= N is roughly ((3/5)*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). (End)

Links

Crossrefs

Cf. A019334, A005596, A000720, A323594.

Programs

  • Mathematica
    pr=27; Select[Prime[Range[300]], MultiplicativeOrder[pr, # ] == #-1 &]
  • PARI
    isA019353(n) = isprime(n) && (n!=3) && znorder(Mod(27,n)) == n-1 \\ Jianing Song, May 12 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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