cp's OEIS Frontend

A211171 Exponent of general linear group GL(n,2).

%I A211171 #40 Jul 29 2022 21:41:00
%S A211171 1,6,84,420,26040,78120,9921240,168661080,24624517680,270869694480,
%T A211171 554470264600560,7208113439807280,59041657185461430480,
%U A211171 2538791258974841510640,383357480105201068106640,98522872387036674503406480,25826982813282567927671981480160
%N A211171 Exponent of general linear group GL(n,2).
%C A211171 a(n) is the smallest integer for which x^a(n) = 1 for any x in GL(n,2).
%H A211171 Alexander Gruber, <a href="/A211171/b211171.txt">Table of n, a(n) for n = 1..100</a>
%H A211171 Gert Almkvist, <a href="https://doi.org/10.1090/S0002-9939-1975-0376713-2">Powers of a matrix with coefficients in a Boolean ring</a>, Proc. Amer. Math. Soc. 53 (1975), 27-31. See u_n.
%H A211171 Eugene Karolinsky and Dmytro Seliutin, <a href="https://arxiv.org/abs/2001.10315">Carmichael numbers for GL(m)</a>, arXiv:2001.10315 [math.NT], 2020; where a(n) is noted as K2(n), see page 1.
%H A211171 MathStackExchange, <a href="http://math.stackexchange.com/a/294524/12952">Exponent of GL(n,q)</a>
%F A211171 a(n) = 2^ceiling(log_2(n)) * Product_{k=1..n} (k-th cyclotomic polynomial evaluated at 2).
%F A211171 a(n) = A034268(n)*A062383(n+1). - _Michel Marcus_, Jul 29 2022
%e A211171 n = 2: GL(2,2) is isomorphic to S3 which has exponent 6 (see: A003418).
%e A211171 n = 3: The set of element orders of GL(3,2) is {1,2,3,4,7} so the exponent is 84.
%e A211171 n = 5: The set of element orders of GL(5,2) is {1,2,3,4,5, 6,7,8,12,14, 15,21,31} so the exponent is 26040 (see: A053651).
%p A211171 with(numtheory):
%p A211171 a:= proc(n) local t; t:= 2^ilog2(n);
%p A211171       `if`(t<n, 2, 1)*t*mul(cyclotomic(k, 2), k=1..n)
%p A211171     end:
%p A211171 seq(a(n), n=1..20);  # _Alois P. Heinz_, Feb 04 2013
%t A211171 f[q_, n_] := With[{p = Sort[Divisors[q]][[2]]},
%t A211171   p^Ceiling[Log[p, n]] Product[Cyclotomic[k, q], {k, n}]]; f[2,#]&/@Range[100]
%o A211171 (Magma)
%o A211171 for n in [1..18] do
%o A211171 Exponent(GL(n,2));
%o A211171 end for;
%o A211171 (PARI) a(n) = 2^ceil(log(n)/log(2))*prod(k=1, n, polcyclo(k, 2)); \\ _Michel Marcus_, Jan 29 2020
%Y A211171 Cf. A003418, A053651.
%Y A211171 Cf. A006951 (number of conjugacy classes in GL(n,2)).
%Y A211171 Cf. A034268, A062383.
%K A211171 nonn
%O A211171 1,2
%A A211171 _Alexander Gruber_, Jan 31 2013