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.

A001915 Primes p such that the congruence 2^x == 3 (mod p) is solvable.

Original entry on oeis.org

2, 5, 11, 13, 19, 23, 29, 37, 47, 53, 59, 61, 67, 71, 83, 97, 101, 107, 131, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 211, 227, 239, 263, 269, 293, 307, 311, 313, 317, 347, 349, 359, 373, 379, 383, 389, 409, 419, 421, 431, 443, 461, 467, 479, 491, 499, 503, 509, 523
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The sequence is known to be infinite [Polya] - thanks to Pieter Moree and Daniel Stefankovic for this comment, Dec 21 2009.

References

  • M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 63.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001916, A050259, A123988.

Programs

  • Maple
    N:= 1000: # to search the first N primes
{2} union select(t -> numtheory[mlog](3,2,p) <> FAIL, {seq(ithprime(n),n=2..N)});
# Robert Israel, Feb 15 2013
  • Mathematica
    Select[Prime[Range[120]], MemberQ[Table[Mod[2^x-3, #], {x, 0, #}], 0]&] (* Jean-François Alcover, Aug 29 2011 *)
Monitor[aaa=Reap[Do[p=Prime[m];sol=MultiplicativeOrder[2,p,{3}];If[IntegerQ[sol],Sow[p]],{m,1000}]],{m}];tmp=Transpose[{1+Range[Length[aaa[[2,1]]]],aaa[[2,1]]}] (* Xianwen Wang, Jul 22 2025 *)
  • PARI
    isok(p) = isprime(p) && sum(k=0, (p-1), Mod(2, p)^k == 3); \\ Michel Marcus, Mar 12 2017
  • PARI
    is(n)=isprime(n) && (n==2 || #znlog(3, Mod(2, n))) \\ Charles R Greathouse IV, Aug 15 2018

Extensions

Better description from Joe K. Crump (joecr(AT)carolina.rr.com), Dec 11 2000
More terms from David W. Wilson, Dec 12 2000

A191609 Primes modulo which the multiplicative orders of 2 and 3 are equal.

Original entry on oeis.org

5, 19, 23, 29, 47, 53, 71, 97, 101, 139, 149, 163, 167, 173, 191, 197, 211, 239, 263, 269, 293, 311, 317, 359, 379, 383, 389, 409, 431, 461, 479, 499, 503, 509, 557, 599, 643, 647, 653, 677, 701, 719, 743, 773, 797, 821, 839, 859, 863, 887, 907, 941, 983
Offset: 1

Views

Author

Max Alekseyev, Jun 08 2011

Keywords

Links

Crossrefs

Cf. A123988, A127437, A127438.

Programs

  • Maple
    select(p -> isprime(p) and numtheory:-order(2,p) = numtheory:-order(3,p), [seq(i,i=5..10000,2)]); # Robert Israel, Jan 24 2024
  • Mathematica
    okQ[p_] := MultiplicativeOrder[2, p] == MultiplicativeOrder[3, p];
Select[Prime[Range[1000]], okQ] (* Jean-François Alcover, Nov 23 2024 *)
  • PARI
    forprime(p=5,10^3, if( znorder(Mod(2,p))==znorder(Mod(3,p)), print1(p,", ");) )

A127437 Duplicate of A001915.

Original entry on oeis.org

2, 5, 11, 13, 19, 23, 29, 37, 47, 53, 59, 61, 67, 71, 83, 97, 101, 107, 131, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 211, 227, 239, 263, 269, 293, 307, 311, 313, 317, 347, 349, 359, 373, 379, 383, 389, 409, 419, 421, 431, 443, 461, 467, 479, 491, 499, 503, 509, 523, 541, 547, 557, 563, 577
Offset: 1

Views

Author

Max Alekseyev, Jan 14 2007

Keywords

Comments

Potential prime divisors of solutions to 2^m == 3 (mod m) (see A050259).
Minimal nonnegative solutions to 2^x == 3 (mod a(n)) are given in A127438.

Crossrefs

Cf. A050259, A123988 (complement in the primes).

Programs

  • PARI
    forprime(p=5,1000, g=znprimroot(p); u=znlog(Mod(2,p),g); v=znlog(Mod(3,p),g); if( v%u==0, print1(p,", "); ))

Extensions

Corrected by Max Alekseyev, Jun 08 2011
Corrected by Arkadiusz Wesolowski, Jan 12 2021
Showing 1-3 of 3 results.

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

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