A218338 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(17) listed in ascending order.

1, 2, 4, 8, 16, 3, 6, 9, 12, 18, 24, 32, 36, 48, 72, 96, 144, 288, 307, 614, 1228, 2456, 4912, 5, 10, 15, 20, 29, 30, 40, 45, 58, 60, 64, 80, 87, 90, 116, 120, 145, 160, 174, 180, 192, 232, 240, 261, 290, 320, 348, 360, 435, 464, 480, 522, 576, 580, 696, 720

Offset: 1

Author

Alois P. Heinz, Oct 26 2012

Keywords

• nonn
• tabf
• look

Examples

			Triangle begins:
      1,      2,      4,      8,      16;
      3,      6,      9,     12,      18,  24, 32, 36, 48, 72, ...
    307,    614,   1228,   2456,    4912;
      5,     10,     15,     20,      29,  30, 40, 45, 58, 60, ...
  88741, 177482, 354964, 709928, 1419856;
		

Links

• Alois P. Heinz, Rows n = 1..19, flattened
• Eric Weisstein's World of Mathematics, Irreducible Polynomial
• Eric Weisstein's World of Mathematics, Polynomial Order

Crossrefs

Column k=7 of A212737.

Column k=1 gives: A218361.

Row lengths are A212957(n,17).

Programs

Maple

with(numtheory):
M:= proc(n) M(n):= divisors(17^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);

Mathematica

M[n_] := M[n] = Divisors[17^n-1] ~Complement~ U[n-1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n-1]];
T[n_] := Sort[M[n]];
Table[T[n], {n, 1, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

Formula

T(n,k) = k-th smallest element of M(n) = {d : d|(17^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.