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A218360 Minimal order of degree-n irreducible polynomials over GF(13).

%I A218360 #15 Feb 16 2025 08:33:18
%S A218360 1,7,9,5,30941,63,5229043,32,27,11,23,45,53,29,4651,64,103,19,
%T A218360 12865927,25,43,161,1381,288,701,371,81,145,1973,31,311,128,207,721,
%U A218360 211,37,1481,90061489,79,41,6740847065723,261,119627,115,181,47,183959,576,1667,101
%N A218360 Minimal order of degree-n irreducible polynomials over GF(13).
%C A218360 a(n) < 13^n.
%H A218360 Max Alekseyev, <a href="/A218360/b218360.txt">Table of n, a(n) for n = 1..416</a>
%H A218360 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IrreduciblePolynomial.html">Irreducible Polynomial</a>
%H A218360 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolynomialOrder.html">Polynomial Order</a>
%F A218360 a(n) = min(M(n)) with M(n) = {d : d|(13^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.
%F A218360 a(n) = A218337(n,1) = A213224(n,6).
%p A218360 with(numtheory):
%p A218360 M:= proc(n) M(n):= divisors(13^n-1) minus U(n-1) end:
%p A218360 U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
%p A218360 a:= n-> min(M(n)[]):
%p A218360 seq(a(n), n=1..33);
%t A218360 M[n_] := M[n] = Divisors[13^n - 1]~Complement~U[n - 1];
%t A218360 U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
%t A218360 a[n_] := Min[M[n]];
%t A218360 Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Oct 24 2022, after _Alois P. Heinz_ *)
%Y A218360 Cf. A213224, A218337.
%K A218360 nonn
%O A218360 1,2
%A A218360 _Alois P. Heinz_, Oct 27 2012