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

1,2

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.

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

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.

Example

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;
  ...

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 *)

Crossrefs

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

Column k=3 of A212737.

Column k=1 gives: A218357.

Sequence in context: A362041 A175334 A128556 * A077661 A077583 A153125

Adjacent sequences: A212482 A212483 A212484 * A212486 A212487 A212488

Keyword

easy,nonn,look,tabf

Author

Boris Putievskiy, Jun 02 2012

Status

approved