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.

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A057481 Numbers n such that x^n + x^11 + 1 is irreducible over GF(2).

Original entry on oeis.org

2, 9, 15, 17, 18, 36, 60, 63, 84, 95, 98, 135, 156, 170, 186, 218, 540, 641, 660, 879, 1388, 1820, 1866, 1943, 2055, 2388, 3423, 3983, 6090, 6713, 9900, 14610, 18330, 18855, 22346, 26180, 32855, 36410, 43911, 44465, 82652, 88764, 131250, 154644, 231420
Offset: 1

Views

Author

Robert G. Wilson v, Sep 27 2000

Keywords

Comments

Any subsequent terms are > 300000.

Links

Crossrefs

Cf. A002475.

Programs

  • PARI
    is(k) = polisirreducible(Mod(1, 2)*(x^k + x^11 + 1)); \\ Jinyuan Wang, Apr 15 2020

Extensions

a(20)-a(28) from Jinyuan Wang, Apr 15 2020
a(29)-a(45) from Lucas A. Brown, Nov 29 2022

A058334 Numbers n such that the trinomial x^n + x + 1 is irreducible over GF(5).

Original entry on oeis.org

0, 1, 2, 3, 7, 18, 22, 27, 31, 78, 94, 115, 171, 402, 438, 507, 1363, 1467, 2263, 2283, 3627, 9247, 9955
Offset: 1

Views

Author

Robert G. Wilson v, Dec 13 2000

Keywords

Comments

No other n < 4400. - Michael Somos, Mar 12 2007
Next term > 10^4. [Joerg Arndt, Mar 02 2016]

Crossrefs

Cf. A002475 (GF(2)), A058857 (GF(7)).

Programs

  • PARI
    isok(n) = polisirreducible(Mod(1, 5)*(x^n + x + 1)); \\ Michel Marcus, Feb 11 2014
  • Sage
    P. = GF(5)[]
for n in range(0, 10000):
       if (x^n+x+1).is_irreducible():
           print(n)
# Joerg Arndt, Mar 02 2016

Extensions

a(1) and a(2) from Eric M. Schmidt, Feb 10 2014
a(22) and a(23) from Joerg Arndt, Mar 02 2016

A133485 Integers k such that the polynomial x^(2k+2) + x + 1 is primitive and irreducible over GF(2).

Original entry on oeis.org

0, 1, 2, 10, 29, 265, 449, 682
Offset: 1

Views

Author

Max Alekseyev, Dec 02 2007, Feb 15 2008

Keywords

Comments

An integer k > 1 belongs to this sequence iff A100730(k) = 2^(2k+3) - 2.
Also, an integer k belongs to this sequence iff 2k+2 belongs to A073639.
The polynomial x^(2k+2) + x + 1 in the definition can be replaced by its reciprocal x^(2k+2) + x^(2k+1) + 1.
(2*a(n)+2) is a subsequence of A002475. - Manfred Scheucher, Aug 17 2015
a(9) >= (A002475(29) - 2)/2 = 5098.

Crossrefs

Cf. A002475, A073639, A100730.

Programs

  • Maple
    select(n -> (Irreduc(x^(2*n+2)+x+1) mod 2) and (Primitive(x^(2*n+2)+x+1) mod 2), [$0..500]); # Robert Israel, Aug 17 2015
  • PARI
    polisprimitive(poli)=np = 2^poldegree(poli)-1; if (type((x^np-1)/poli) != "t_POL", return (0)); forstep(k=np-1, 1, -1, if (type((x^k-1)/poli) == "t_POL", return (0));); return(1);
lista(nn) = {for (n=0, nn, poli = Mod(1,2)*(x^(2*n+2)+x+1); if (polisirreducible(poli) && polisprimitive(poli), print1(n, ", ")););} \\ Michel Marcus, May 27 2013
  • Sage
    def is_primitive(p):
    d = 2^(p.degree())-1
    if not p.divides(x^d-1): return False
    for k in (d//q for q in d.prime_factors()):
        if p.divides(x^k-1): return False
    return True
P. = GF(2)[]
for n in range(1,1000):
    p = x^(2*n+2)+x+1
    if p.is_irreducible() and is_primitive(p):
        print(n)
# Modification of the A002475 Script by Ruperto Corso
# Manfred Scheucher, Aug 17 2015

Extensions

a(2)=1 inserted by Michel Marcus, May 29 2013
Previous Showing 21-23 of 23 results.

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

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