OFFSET
1,2
COMMENTS
a(n) is the smallest integer for which x^a(n) = 1 for any x in GL(n,2).
LINKS
Alexander Gruber, Table of n, a(n) for n = 1..100
Gert Almkvist, Powers of a matrix with coefficients in a Boolean ring, Proc. Amer. Math. Soc. 53 (1975), 27-31. See u_n.
Eugene Karolinsky and Dmytro Seliutin, Carmichael numbers for GL(m), arXiv:2001.10315 [math.NT], 2020; where a(n) is noted as K2(n), see page 1.
MathStackExchange, Exponent of GL(n,q)
FORMULA
a(n) = 2^ceiling(log_2(n)) * Product_{k=1..n} (k-th cyclotomic polynomial evaluated at 2).
EXAMPLE
MAPLE
with(numtheory):
a:= proc(n) local t; t:= 2^ilog2(n);
`if`(t<n, 2, 1)*t*mul(cyclotomic(k, 2), k=1..n)
end:
seq(a(n), n=1..20); # Alois P. Heinz, Feb 04 2013
MATHEMATICA
f[q_, n_] := With[{p = Sort[Divisors[q]][[2]]},
p^Ceiling[Log[p, n]] Product[Cyclotomic[k, q], {k, n}]]; f[2, #]&/@Range[100]
PROG
(Magma)
for n in [1..18] do
Exponent(GL(n, 2));
end for;
(PARI) a(n) = 2^ceil(log(n)/log(2))*prod(k=1, n, polcyclo(k, 2)); \\ Michel Marcus, Jan 29 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexander Gruber, Jan 31 2013
STATUS
approved