A212486 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(7) listed in ascending order.

1, 2, 3, 6, 4, 8, 12, 16, 24, 48, 9, 18, 19, 38, 57, 114, 171, 342, 5, 10, 15, 20, 25, 30, 32, 40, 50, 60, 75, 80, 96, 100, 120, 150, 160, 200, 240, 300, 400, 480, 600, 800, 1200, 2400, 2801, 5602, 8403, 16806, 36, 43, 72, 76, 86, 129, 144, 152, 172, 228, 258

Offset: 1

Author

Boris Putievskiy, Jun 02 2012

Keywords

• easy
• nonn
• tabf

Comments

The elements m of row n, are also solutions to the equation: multiplicative order of 7 mod m = n, with gcd(m,7) = 1, cf. A053450.

Examples

			Triangle T(n,k) begins:
  1,  2,  3,  6;
  4,  8, 12, 16, 24,  48;
  9, 18, 19, 38, 57, 114, 171, 342;
  5, 10, 15, 20, 25,  30,  32,  40, 50, 60, 75, 80, 96, 100, 120, 150, 160, 200, 240, 300, 400, 480, 600, 800, 1200, 2400;
  ...
		

References

• R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 560-562.
• V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28

Links

• Boris Putievskiy, Table of n, a(n) for n = 1..102
• Eric Weisstein's World of Mathematics, Irreducible Polynomial
• Gang Xiao, (computes the order of an irreducible polynomial over a finite field GF(p))

Crossrefs

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

Column k=4 of A212737.

Column k=1 gives: A218358.

Programs

Maple

with(numtheory):
M:= proc(n) option remember;
      `if`(n=1, {1, 2, 3, 6}, divisors(7^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..7);

Mathematica

M[n_] := M[n] = If[n == 1, {1, 2, 3, 6}, Divisors[7^n - 1] ~Complement~ U[n - 1]];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
T[n_] := Sort[M[n]];
Table[T[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Sep 24 2022, from Maple code *)

Formula

T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (7^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2,3,6}.

|M(n)| = Sum_{d|n} mu(n/d)*tau(7^d-1) = A059889(n).