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A212906 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(3) listed in ascending order.

%I A212906 #47 Feb 16 2025 08:33:17
%S A212906 1,2,4,8,13,26,5,10,16,20,40,80,11,22,121,242,7,14,28,52,56,91,104,
%T A212906 182,364,728,1093,2186,32,41,82,160,164,205,328,410,656,820,1312,1640,
%U A212906 3280,6560,757,1514,9841,19682,44,61,88,122,244,484,488,671,968,1342
%N A212906 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(3) listed in ascending order.
%C A212906 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 A212906 R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 555-557.
%D A212906 V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28
%H A212906 Alois P. Heinz, <a href="/A212906/b212906.txt">Rows n = 1..47, flattened</a> (first 13 rows from Boris Putievskiy)
%H A212906 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IrreduciblePolynomial.html">Irreducible Polynomial</a>
%H A212906 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 A212906 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 A212906 |M(n)| = Sum_{d|n} mu(n/d)*tau(3^d-1) = A059885(n).
%e A212906 Triangle T(n,k) begins:
%e A212906 1,   2;
%e A212906 4,   8;
%e A212906 13, 26;
%e A212906 5,  10,  16,  20, 40, 80;
%e A212906 11, 22, 121, 242;
%e A212906 7,  14,  28,  52, 56, 91, 104, 182, 364, 728;
%p A212906 with(numtheory):
%p A212906 M:= proc(n) option remember;
%p A212906       divisors(3^n-1) minus U(n-1)
%p A212906     end:
%p A212906 U:= proc(n) option remember;
%p A212906       `if`(n=0, {}, M(n) union U(n-1))
%p A212906     end:
%p A212906 T:= n-> sort([M(n)[]])[]:
%p A212906 seq(T(n), n=1..15);  # _Alois P. Heinz_, Jun 02 2012
%t A212906 M[n_] := M[n] = Divisors[3^n - 1] ~Complement~ U[n - 1];
%t A212906 U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
%t A212906 T[n_] := Sort[M[n]]; Array[T, 15] // Flatten (* _Jean-François Alcover_, Jun 10 2018, after _Alois P. Heinz_ *)
%Y A212906 Cf. A053446, A059912, A059885, A058944, A059499, A059886-A059892.
%Y A212906 Column k=2 of A212737.
%Y A212906 Column k=1 gives: A218356.
%K A212906 easy,nonn,look,tabf
%O A212906 1,2
%A A212906 _Boris Putievskiy_, May 29 2012