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A212906 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(3) listed in ascending order. 7
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 (list; graph; refs; listen; history; text; internal format)
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

Boris Putievskiy and 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 A043777 A043781

Adjacent sequences:  A212903 A212904 A212905 * A212907 A212908 A212909

KEYWORD

easy,nonn,look,tabf,changed

AUTHOR

Boris Putievskiy, May 29 2012

STATUS

approved

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Last modified June 19 16:05 EDT 2018. Contains 305594 sequences. (Running on oeis4.)