A057764 Triangle T(n,k) = number of nonzero elements of multiplicative order k in Galois field GF(2^n) (n >= 1, 1 <= k <= 2^n-1).

1, 1, 0, 2, 1, 0, 0, 0, 0, 0, 6, 1, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 1, 0, 2, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36

Offset: 1

Author

N. J. A. Sloane, Nov 01 2000

Keywords

• nonn
• easy
• nice
• tabf

Examples

			Table begins:
  1;
  1, 0, 2;
  1, 0, 0, 0, 0, 0, 6;
  ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..16369 (rows 1 to 13, flattened)

Crossrefs

Cf. A000010, A000225, A002326, A053287.

Programs

Magma

{* Order(g) : g in GF(2^6) | g ne 0 *};

Maple

f:= proc(n,k) if 2^n-1 mod k = 0 then numtheory:-phi(k) else 0 fi end proc:
seq(seq(f(n,k),k=1..2^n-1), n=1..10); # Robert Israel, Jul 21 2016

Mathematica

T[n_, k_] := If[Divisible[2^n - 1, k], EulerPhi[k], 0];
Table[T[n, k], {n, 1, 10}, {k, 1, 2^n - 1}] // Flatten (* Jean-François Alcover, Feb 07 2023, after Robert Israel *)

Formula

From Robert Israel, Jul 21 2016: (Start)

T(n,k) = A000010(k) if k is a divisor of 2^n-1, otherwise 0.

Sum_{k=1..2^n-1} T(n,k) = 2^n-1 = A000225(n).

G.f. as triangle: g(x,y) = Sum_{j>=0} x^A002326(j)*A000010(2j+1)*y^(2j+1)/(1-x^A002326(j)). (End)

Extensions

T(6,21) corrected by Robert Israel, Jul 21 2016