A218357 Minimal order of degree-n irreducible polynomials over GF(5).

1, 3, 31, 13, 11, 7, 19531, 32, 19, 33, 12207031, 91, 305175781, 29, 181, 17, 409, 27, 191, 41, 379, 23, 8971, 224, 101, 5227, 109, 377, 59, 61, 1861, 128, 199, 1227, 211, 37, 149, 573, 79, 241, 2238236249, 43, 1644512641, 89, 209, 47, 177635683940025046467781066894531

Offset

1,2

Comments

a(n) < 5^n.

a(n) <= A143665(n). For prime n, a(n) = A143665(n). - Max Alekseyev, Apr 30 2022

Links

Max Alekseyev, Table of n, a(n) for n = 1..520

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|(5^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A212485(n,1) = A213224(n,3).

Maple

with(numtheory):
M:= proc(n) M(n):= divisors(5^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..47);

Mathematica

M[n_] := M[n] = Divisors[5^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, 47}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

Crossrefs

Cf. A212485, A213224.

Sequence in context: A322777 A089281 A212729 * A090543 A215946 A139090

Adjacent sequences: A218354 A218355 A218356 * A218358 A218359 A218360

Keyword

nonn

Author

Alois P. Heinz, Oct 27 2012

Status

approved