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

%I A218363 #17 Feb 16 2025 08:33:18
%S A218363 1,3,7,5,292561,9,29,64,19,31,121,35,47691619,71,2047927,17,103,27,
%T A218363 2129,25,43,363,461,448,6551,143074857,4591,145,233,151,
%U A218363 40888990028603,193,67,239,8484269,73,1925658337781,6387,333841333,1600,83,129,173,605,5558659
%N A218363 Minimal order of degree-n irreducible polynomials over GF(23).
%C A218363 a(n) < 23^n.
%H A218363 Max Alekseyev, <a href="/A218363/b218363.txt">Table of n, a(n) for n = 1..306</a>
%H A218363 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IrreduciblePolynomial.html">Irreducible Polynomial</a>
%H A218363 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolynomialOrder.html">Polynomial Order</a>
%F A218363 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) = {}.
%F A218363 a(n) = A218340(n,1) = A213224(n,9).
%p A218363 with(numtheory):
%p A218363 M:= proc(n) M(n):= divisors(23^n-1) minus U(n-1) end:
%p A218363 U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
%p A218363 a:= n-> min(M(n)[]):
%p A218363 seq(a(n), n=1..30);
%t A218363 M[n_] := M[n] = Divisors[23^n - 1]~Complement~U[n - 1];
%t A218363 U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
%t A218363 a[n_] := Min[M[n]];
%t A218363 Table[Print[n, " ", a[n]]; a[n], {n, 1, 45}] (* _Jean-François Alcover_, Oct 24 2022, after _Alois P. Heinz_ *)
%Y A218363 Cf. A213224, A218340.
%K A218363 nonn
%O A218363 1,2
%A A218363 _Alois P. Heinz_, Oct 27 2012