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.

A001133 Primes p such that the multiplicative order of 2 modulo p is (p-1)/3.

Original entry on oeis.org

43, 109, 157, 229, 277, 283, 307, 499, 643, 691, 733, 739, 811, 997, 1021, 1051, 1069, 1093, 1459, 1579, 1597, 1627, 1699, 1723, 1789, 1933, 2179, 2203, 2251, 2341, 2347, 2749, 2917, 3163, 3181, 3229, 3259, 3373, 4027, 4339, 4549, 4597, 4651, 4909, 5101, 5197, 5323, 5413, 5437, 5653, 6037
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 59.
  • 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. A040028, A014752, A059914.
Cf. also A001134, A001135, A001136, A115586, A115591.

Programs

  • Magma
    [ p: p in PrimesUpTo(4597) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,3) where R is ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008
  • Mathematica
    Reap[For[p = 2, p < 10^4, p = NextPrime[p], If[MultiplicativeOrder[2, p] == (p-1)/3, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 10 2015 *)
  • PARI
    forprime(p=3,10^4,if(znorder(Mod(2,p))==(p-1)/3,print1(p,", "))); \\ Joerg Arndt, May 17 2013

Extensions

More terms and better definition from Don Reble, Mar 11 2006

A001134 Primes p such that the multiplicative order of 2 modulo p is (p-1)/4.

Original entry on oeis.org

113, 281, 353, 577, 593, 617, 1033, 1049, 1097, 1153, 1193, 1201, 1481, 1601, 1889, 2129, 2273, 2393, 2473, 3049, 3089, 3137, 3217, 3313, 3529, 3673, 3833, 4001, 4217, 4289, 4457, 4801, 4817, 4937, 5233, 5393, 5881, 6121, 6521, 6569, 6761, 6793, 6841, 7129, 7481, 7577, 7793, 7817, 7841, 8209
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The multiplicative order of x modulo y is the smallest positive number m such that x^m is congruent to 1 mod y.

References

  • M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 59.
  • 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. A001133, A001135, A001136, A115586, A115591.

Programs

  • Magma
    [ p: p in PrimesUpTo(6761) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,4) where R is ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008
  • Mathematica
    Reap[For[p = 2, p <= 6761, p = NextPrime[p], If[ MultiplicativeOrder[2, p] == (p-1)/4, Sow[p]]]][[2, 1]] (* Jean-François Alcover, May 17 2013 *)
  • PARI
    forprime(p=3,10^4,if(znorder(Mod(2,p))==(p-1)/4,print1(p,", "))); \\ Joerg Arndt, May 17 2013
  • PARI
    oddres(n)=n>>valuation(n, 2)
cyc(d)=my(k=1, t=1,y=(d-5)/(2*3)+1); while((t=oddres(t+d))>1 && k<=y, k++); k
forstep(n=1, 241537, [16,8], if(cyc(n)==n>>3,print1(n", "))) ; \\ Charles R Greathouse IV, May 18 2013

Extensions

More terms and better definition from Don Reble, Mar 11 2006

A216838 Odd primes for which 2 is not a primitive root.

Original entry on oeis.org

7, 17, 23, 31, 41, 43, 47, 71, 73, 79, 89, 97, 103, 109, 113, 127, 137, 151, 157, 167, 191, 193, 199, 223, 229, 233, 239, 241, 251, 257, 263, 271, 277, 281, 283, 307, 311, 313, 331, 337, 353, 359, 367, 383, 397, 401, 409, 431, 433, 439, 449, 457, 463, 479
Offset: 1

Views

Author

V. Raman, Sep 17 2012

Keywords

Comments

Alternately, for these primes p, the polynomial (x^p+1)/(x+1) is reducible over GF(2).
The prime p belongs to this sequence if and only if A002326((p-1)/2) != (p-1). If A002326((p-1)/2) = (p-1), then the prime p belongs to the sequence A001122. - V. Raman, Dec 01 2012
The only primitive root modulo 2 is 1. See A060749. Hence 2 should be added to this sequence in order to obtain the complement of A001122. - Wolfdieter Lang, May 19 2014

Links

Crossrefs

Cf. A002326 (multiplicative order of 2 mod 2n+1)
Cf. A001122 (Primes for which 2 is a primitive root)
Cf. A115586 (Primes for which 2 is neither a primitive root nor a quadratic residue).

Programs

  • Maple
    select(t -> isprime(t) and numtheory[order](2,t) <> t-1, [seq](2*i+1,i=1..1000)); # Robert Israel, May 20 2014
  • Mathematica
    Select[Prime[Range[2, 100]], PrimitiveRoot[#] =!= 2 &] (* T. D. Noe, Sep 19 2012 *)
  • PARI
    forprime(p=3, 1000, if(znorder(Mod(2,p))!=p-1, print(p)))
  • PARI
    forprime(p=3, 1000, if(factormod((x^p+1)/(x+1), 2, 1)[1, 1]!=(p-1), print(p)))

Extensions

Name corrected by Wolfdieter Lang, May 19 2014

A001135 Primes p such that the multiplicative order of 2 modulo p is (p-1)/5.

Original entry on oeis.org

251, 571, 971, 1181, 1811, 2011, 2381, 2411, 3221, 3251, 3301, 3821, 4211, 4861, 4931, 5021, 5381, 5861, 6221, 6571, 6581, 8461, 8501, 9091, 9461, 10061, 10211, 10781, 11251, 11701, 11941, 12541, 13171, 13381, 13421, 13781, 14251, 15541, 16091, 16141, 16451, 16661, 16691, 16811, 17291
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 59.
  • 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. A001133, A001134, A001136, A115586, A115591.

Programs

  • Magma
    [ p: p in PrimesUpTo(15541) | r eq 1 and Order(R!2) eq q where q,r is Quotrem(p,5) where R is ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008
  • Maple
    q:= p-> isprime(p) and numtheory[order](2, p)=(p-1)/5:
select(q, [$2..20000])[];  # Alois P. Heinz, Dec 12 2023
  • Mathematica
    Reap[For[p = 2, p <= 18000, p = NextPrime[p], If[ MultiplicativeOrder[2, p] == (p-1)/5, Sow[p]]]][[2, 1]] (* James C. McMahon, Dec 12 2023 *)
  • PARI
    forprime(p=3,10^5,if(znorder(Mod(2,p))==(p-1)/5,print1(p,", "))); \\ Joerg Arndt, May 17 2013

Extensions

More terms and better definition from Don Reble, Mar 11 2006
Showing 1-4 of 4 results.

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

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