A059912 Triangle T(n,k) of orders of n degree irreducible polynomials over GF(2) listed in ascending order, k=1..A059499(n).

1, 3, 7, 5, 15, 31, 9, 21, 63, 127, 17, 51, 85, 255, 73, 511, 11, 33, 93, 341, 1023, 23, 89, 2047, 13, 35, 39, 45, 65, 91, 105, 117, 195, 273, 315, 455, 585, 819, 1365, 4095, 8191, 43, 129, 381, 5461, 16383, 151, 217, 1057, 4681, 32767, 257, 771, 1285, 3855

Offset

1,2

Comments

A permutation of the odd positive numbers; namely, order each odd number d by the multiplicative order of 2 modulo d (in case of a tie, smaller d go first). - Jeppe Stig Nielsen, Feb 13 2020

Links

Alois P. Heinz, Rows n = 1..71, flattened

Eric Weisstein's World of Mathematics, Irreducible Polynomial

XIAO Gang, Polynomial order: computes the order of an irreducible polynomial over a finite field GF(p), WIMS.

Formula

T(n,k) = k-th smallest element of M(n) = {d : d|(2^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}. - Alois P. Heinz, Jun 01 2012

Example

There are 18 (cf. A001037) irreducible polynomials of degree 7 over GF(2) which all have order 127.
Triangle T(n,k) begins:
    1;
    3;
    7;
    5,   15;
   31;
    9,   21,  63;
  127;
   17,   51,  85, 255;
   73,  511;
   11,   33,  93, 341, 1023;
  ...

Maple

with(numtheory):
M:= proc(n) option remember;
      divisors(2^n-1) minus U(n-1)
    end:
U:= proc(n) option remember;
      `if`(n=0, {}, M(n) union U(n-1))
    end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..20);  # Alois P. Heinz, May 31 2012

Mathematica

m[n_] := m[n] = Complement[ Divisors[2^n - 1], u[n - 1]]; u[0] = {}; u[n_] := u[n] = Union[ m[n], u[n - 1]]; t[n_, k_] := m[n][[k]]; Flatten[ Table[t[n, k], {n, 1, 16}, {k, 1, Length[ m[n] ]}]] (* Jean-François Alcover, Jun 14 2012, after Alois P. Heinz *)

Prog

(PARI) maxDegree=26; for(n=1, maxDegree, forstep(d=1, 2^n, 2, znorder(Mod(2, d))==n&&print1(d, ", "))) \\ inefficient, Jeppe Stig Nielsen, Feb 13 2020

Crossrefs

Cf. A058943, A059478, A059499, A001037, A059913.

Cf. A212906, A212485, A212486.

Column k=1 of A212737.

Column k=1 gives: A212953.

Last elements of rows give: A000225.

Cf. A108974.

Sequence in context: A369277 A324821 A345401 * A354008 A115765 A282598

Adjacent sequences: A059909 A059910 A059911 * A059913 A059914 A059915

Keyword

easy,nonn,look,tabf

Author

Vladeta Jovovic, Feb 09 2001

Status

approved