A218363 Minimal order of degree-n irreducible polynomials over GF(23).

1, 3, 7, 5, 292561, 9, 29, 64, 19, 31, 121, 35, 47691619, 71, 2047927, 17, 103, 27, 2129, 25, 43, 363, 461, 448, 6551, 143074857, 4591, 145, 233, 151, 40888990028603, 193, 67, 239, 8484269, 73, 1925658337781, 6387, 333841333, 1600, 83, 129, 173, 605, 5558659

Offset

1,2

Comments

a(n) < 23^n.

Links

Max Alekseyev, Table of n, a(n) for n = 1..306

Eric Weisstein's World of Mathematics, Irreducible Polynomial

Eric Weisstein's World of Mathematics, Polynomial Order

Formula

a(n) = min(M(n)) with M(n) = {d : d|(23^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A218340(n,1) = A213224(n,9).

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:
a:= n-> min(M(n)[]):
seq(a(n), n=1..30);

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]];
a[n_] := Min[M[n]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 45}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)

Crossrefs

Cf. A213224, A218340.

Sequence in context: A277207 A001663 A085052 * A200611 A016666 A318352

Adjacent sequences: A218360 A218361 A218362 * A218364 A218365 A218366

Keyword

nonn

Author

Alois P. Heinz, Oct 27 2012

Status

approved