A218340 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(23) listed in ascending order.

1, 2, 11, 22, 3, 4, 6, 8, 12, 16, 24, 33, 44, 48, 66, 88, 132, 176, 264, 528, 7, 14, 77, 79, 154, 158, 553, 869, 1106, 1738, 6083, 12166, 5, 10, 15, 20, 30, 32, 40, 53, 55, 60, 80, 96, 106, 110, 120, 159, 160, 165, 212, 220, 240, 265, 318, 330, 352, 424, 440

Offset

1,2

Links

Alois P. Heinz, Rows n = 1..17, flattened

Eric Weisstein's World of Mathematics, Irreducible Polynomial

Eric Weisstein's World of Mathematics, Polynomial Order

Formula

T(n,k) = k-th smallest element of M(n) = {d : d|(23^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

Example

Triangle begins:
       1,      2,      11,      22;
       3,      4,       6,       8,  12,  16,  24,  33,   44, ...
       7,     14,      77,      79, 154, 158, 553, 869, 1106, ...
       5,     10,      15,      20,  30,  32,  40,  53,   55, ...
  292561, 585122, 3218171, 6436342;
  ...

Maple

with(numtheory):
M:= proc(n) M(n):= divisors(23^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[23^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 *)

Crossrefs

Column k=9 of A212737.

Column k=1 gives: A218363.

Row lengths are A212957(n,23).

Sequence in context: A085652 A111090 A111081 * A018491 A031010 A161708

Adjacent sequences: A218337 A218338 A218339 * A218341 A218342 A218343

Keyword

nonn,tabf,look

Author

Alois P. Heinz, Oct 26 2012

Status

approved