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%I #46 Oct 27 2023 21:40:19
%S 1,2,4,8,13,26,5,10,16,20,40,80,11,22,121,242,7,14,28,52,56,91,104,
%T 182,364,728,1093,2186,32,41,82,160,164,205,328,410,656,820,1312,1640,
%U 3280,6560,757,1514,9841,19682,44,61,88,122,244,484,488,671,968,1342
%N Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(3) listed in ascending order.
%C The elements m of row n, are also solutions to the equation: multiplicative order of 3 mod m = n, with gcd(m,3) = 1, cf. A053446.
%D R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 555-557.
%D V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28
%H Alois P. Heinz, <a href="/A212906/b212906.txt">Rows n = 1..47, flattened</a> (first 13 rows from Boris Putievskiy)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IrreduciblePolynomial.html">Irreducible Polynomial</a>
%H XIAO, <a href="http://wims.unice.fr/wims/wims.cgi?module=tool/algebra/polyorder.en">Polynomial order</a> (computes the order of an irreducible polynomial over a finite field GF(p))
%F T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (3^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2}.
%F |M(n)| = Sum_{d|n} mu(n/d)*tau(3^d-1) = A059885(n).
%e Triangle T(n,k) begins:
%e 1, 2;
%e 4, 8;
%e 13, 26;
%e 5, 10, 16, 20, 40, 80;
%e 11, 22, 121, 242;
%e 7, 14, 28, 52, 56, 91, 104, 182, 364, 728;
%p with(numtheory):
%p M:= proc(n) option remember;
%p divisors(3^n-1) minus U(n-1)
%p end:
%p U:= proc(n) option remember;
%p `if`(n=0, {}, M(n) union U(n-1))
%p end:
%p T:= n-> sort([M(n)[]])[]:
%p seq(T(n), n=1..15); # _Alois P. Heinz_, Jun 02 2012
%t M[n_] := M[n] = Divisors[3^n - 1] ~Complement~ U[n - 1];
%t U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
%t T[n_] := Sort[M[n]]; Array[T, 15] // Flatten (* _Jean-François Alcover_, Jun 10 2018, after _Alois P. Heinz_ *)
%Y Cf. A053446, A059912, A059885, A058944, A059499, A059886-A059892.
%Y Column k=2 of A212737.
%Y Column k=1 gives: A218356.
%K easy,nonn,look,tabf
%O 1,2
%A _Boris Putievskiy_, May 29 2012
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