A212485 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(5) listed in ascending order.

1, 2, 4, 3, 6, 8, 12, 24, 31, 62, 124, 13, 16, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624, 11, 22, 44, 71, 142, 284, 781, 1562, 3124, 7, 9, 14, 18, 21, 28, 36, 42, 56, 63, 72, 84, 93, 126, 168, 186, 217, 248, 252, 279, 372, 434, 504, 558, 651, 744, 868, 1116

Offset: 1

Author

Boris Putievskiy, Jun 02 2012

Keywords

• easy
• nonn
• look
• tabf

Comments

The elements m of row n, are also solutions to the equation: multiplicative order of 5 mod m = n, with gcd(m,5) = 1, cf. A050977.

Examples

			Triangle T(n,k) begins:
   1,  2,   4;
   3,  6,   8, 12,  24;
  31, 62, 124;
  13, 16,  26, 39,  48,  52,  78,  104,  156, 208, 312, 624;
  11, 22,  44, 71, 142, 284, 781, 1562, 3124;
  ...
		

References

• R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 557-560.

Links

• Alois P. Heinz, Rows n = 1..32, flattened
• V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28

Crossrefs

Cf. A212906, A059912, A058944, A059499, A059886-A059892.

Column k=3 of A212737.

Column k=1 gives: A218357.

Programs

Maple

with(numtheory):
M:= proc(n) option remember;
      `if`(n=1, {1, 2, 4}, divisors(5^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..8);

Mathematica

M[n_] := M[n] = If[n == 1, {1, 2, 4}, Divisors[5^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, 8] // Flatten (* Jean-François Alcover, Jun 10 2018, from Maple *)

Formula

T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (5^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2,4}.

|M(n)| = Sum_{d|n} mu(n/d)*tau(5^d-1) = A059887.