A346019 Number of n X n invertible matrices over GF(2) that have order 2^n-1.

1, 2, 48, 2688, 1935360, 1919877120, 23222833643520, 335564785519165440, 65717007596073359769600, 21492090164219831579049984000, 66041307304745851496871108594892800, 226523509196861965428709270554756199219200, 16622838761287803491875715175557341313583022080000

Offset: 1

Author

Geoffrey Critzer, Jul 01 2021

Keywords

• nonn

Comments

Equivalently, a(n) is the number of n X n matrices over GF(2) whose characteristic polynomial is primitive.

2^n - 1 is the greatest order that a matrix in the general linear group GL_n(F_2) can have.

Links

• M. R. Darafsheh, Order of elements in the groups related to the general linear group, Finite fields and their applications, 11 (2005), 738-747.

Crossrefs

Cf. A000010, A011260, A002884, A000225, A345463.

Programs

Maple

a:= n-> mul(2^n-2^i, i=0..n-1)*numtheory[phi](2^n-1)/((2^n-1)*n):
seq(a(n), n=1..14);  # Alois P. Heinz, Jul 01 2021

Mathematica

nn = 13; Table[EulerPhi[2^n - 1]/n, {n, 1, nn}]* Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 1, nn}]/Table[2^n - 1, {n, 1, nn}]

Formula

a(n) = A011260(n) * A002884(n)/A000225(n).