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

1, 2, 4, 8, 13, 26, 5, 10, 16, 20, 40, 80, 11, 22, 121, 242, 7, 14, 28, 52, 56, 91, 104, 182, 364, 728, 1093, 2186, 32, 41, 82, 160, 164, 205, 328, 410, 656, 820, 1312, 1640, 3280, 6560, 757, 1514, 9841, 19682, 44, 61, 88, 122, 244, 484, 488, 671, 968, 1342

Offset

1,2

Comments

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.

References

R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 555-557.

V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28

Links

Alois P. Heinz, Rows n = 1..47, flattened (first 13 rows from Boris Putievskiy)

Eric Weisstein's World of Mathematics, Irreducible Polynomial

XIAO, Polynomial order (computes the order of an irreducible polynomial over a finite field GF(p))

Formula

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}.

|M(n)| = Sum_{d|n} mu(n/d)*tau(3^d-1) = A059885(n).

Example

Triangle T(n,k) begins:
1,   2;
4,   8;
13, 26;
5,  10,  16,  20, 40, 80;
11, 22, 121, 242;
7,  14,  28,  52, 56, 91, 104, 182, 364, 728;

Maple

with(numtheory):
M:= proc(n) option remember;
      divisors(3^n-1) minus U(n-1)
    end:
U:= proc(n) option remember;
      `if`(n=0, {}, M(n) union U(n-1))
    end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..15);  # Alois P. Heinz, Jun 02 2012

Mathematica

M[n_] := M[n] = Divisors[3^n - 1] ~Complement~ U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
T[n_] := Sort[M[n]]; Array[T, 15] // Flatten (* Jean-François Alcover, Jun 10 2018, after Alois P. Heinz *)

Crossrefs

Cf. A053446, A059912, A059885, A058944, A059499, A059886-A059892.

Column k=2 of A212737.

Column k=1 gives: A218356.

Sequence in context: A248876 A102704 A196720 * A043774 A361818 A043777

Adjacent sequences: A212903 A212904 A212905 * A212907 A212908 A212909

Keyword

easy,nonn,look,tabf

Author

Boris Putievskiy, May 29 2012

Status

approved