A059478 -id:A059478 - cp's OEIS frontend

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

Vladeta Jovovic, Feb 09 2001

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

			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;
  ...

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.

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.

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

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 *)

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

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

A059913 Triangle T(n,k) of numbers of n degree irreducible polynomials over GF(2) which have order A059912(n,k), k=1..A059499(n).

Original entry on oeis.org

2, 1, 2, 1, 2, 6, 1, 2, 6, 18, 2, 4, 8, 16, 8, 48, 1, 2, 6, 30, 60, 2, 8, 176, 1, 2, 2, 2, 4, 6, 4, 6, 8, 12, 12, 24, 24, 36, 48, 144, 630, 3, 6, 18, 378, 756, 10, 12, 60, 300, 1800, 16, 32, 64, 128, 256, 512, 1024, 2048, 7710, 1, 1, 2, 6, 6, 6, 8, 12, 18, 24

Offset: 1

Vladeta Jovovic, Feb 09 2001

Row sums give A001037.

			There are 9 (cf. A001037) irreducible polynomials of degree 6 over GF(2): 1 of order 9, 2 of order 21 and 6 of order 63 (cf. A059912).
Triangle T(n,k) begins:
  2;
  1;
  2;
  1,  2;
  6;
  1,  2,   6;
  18;
  2,  4,   8, 16;
  8, 48;
  1,  2,   6, 30, 60;
  2,  8, 176;
  ...

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

Cf. A058943, A059478, A059499, A001037, A059912, A000010.

Prepend[Table[Map[EulerPhi[#]/n &, Complement[Divisors[2^n - 1],Union[Flatten[Table[Divisors[2^k - 1], {k, 1, n - 1}]]]]], {n, 2,20}], {2}] // Grid (* Geoffrey Critzer, Dec 02 2019 *)

T(n,k) = phi(A059912(n,k))/n, where phi = Euler totient function A000010.

cp's OEIS Frontend

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

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