A335804 Number of n X n matrices over GF(2) with minimal polynomial x^n - 1.

1, 3, 56, 2520, 666624, 839946240, 3343877406720, 41781748196966400, 3701652434038082764800, 763416952708225267547504640, 750836199529096452135514747699200

Offset: 1

Author

Christof Beierle, Jun 24 2020

Keywords

• nonn

Comments

a(n) is the size of the conjugacy class in GL(n,GF(2)) corresponding to the companion matrix of x^n - 1. It can be given by the number of n X n invertible matrices over GF(2) divided by the number of n X n circulant invertible matrices over GF(2) (i.e., the centralizer of the companion matrix of x^n - 1).

If m is odd, x^m-1 has no multiple roots so that every matrix with characteristic polynomial x^m-1 also has x^m-1 as its minimal polynomial. Hence, a(m) = A089035(m). - Geoffrey Critzer, Jul 24 2025

Crossrefs

Cf. A002884, A003473, A027362, A089035.

Formula

a(n) = A002884(n) / A003473(n). If n is an odd prime, then a(n) = A089035(n).