A218362 Minimal order of degree-n irreducible polynomials over GF(19).

1, 4, 27, 16, 151, 7, 701, 17, 81, 11, 104281, 13, 599, 197, 31, 64, 3044803, 199, 109912203092239643840221, 176, 18927, 23, 277, 119, 101, 131, 243, 29, 59, 61, 243270318891483838103593381595151809701, 97, 67, 12179212, 71, 37, 149, 108301, 79, 41, 10654507

Offset: 1

Author

Alois P. Heinz, Oct 27 2012

Keywords

• nonn

Comments

a(n) < 19^n.

Links

• Max Alekseyev, Table of n, a(n) for n = 1..238
• Eric Weisstein's World of Mathematics, Irreducible Polynomial
• Eric Weisstein's World of Mathematics, Polynomial Order

Crossrefs

Cf. A213224, A218339.

Programs

Maple

with(numtheory):
M:= proc(n) M(n):= divisors(19^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..28);

Mathematica

M[n_] := M[n] = Divisors[19^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[a[n], {n, 1, 41}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)

Formula

a(n) = min(M(n)) with M(n) = {d : d|(19^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A218339(n,1) = A213224(n,8).