A005417 Maximal period of an n-stage shift register. (Formerly M1587)

2, 6, 12, 30, 60, 120, 210, 420, 840, 1260, 2520, 2520, 5040, 9240, 13860, 27720, 32760, 55440, 65520, 120120, 180180, 360360, 360360, 720720, 720720, 942480, 1113840

Offset

0,1

Comments

Maximal order of an element of finite order in GL(2n, Z) or GL(2n+1, Z).

a(n) is the max of the first n numbers in A080742.

References

H. Lüneburg, Galoisfelder, Kreisteilungskörper und Schieberegisterfolgen. B. I. Wissenschaftsverlag, Mannheim, 1979.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..26.

Benjamin Grossmann, What is the largest (finite) order of an element of GL(10,Q)?

J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186.

Formula

a(n) = max m such that A067240(m) <= 2n + 1. E.g., a(2) = 12 since 12 is largest m such that A067240(m) <= 5.

Mathematica

(* b, c = a080737 *)
nmax = 26;
kmax = 1200000; (* kmax increased by 100000 until results do not change *)
b[1] = b[2] = 0; b[p_?PrimeQ] := b[p] = p-1; b[k_] := b[k] = If[Length[f = FactorInteger[k]]==1, EulerPhi[k], Total[b /@ (f[[All, 1]]^f[[All, 2]])] ];
orders = Table[{k, b[k]}, {k, 1, kmax}];
c[0] = 2; c[n_] := c[n] = Select[orders, 2n-1 <= #[[2]] <= 2n&][[-1, 1]];
a[n_] := Table[c[m], {m, 0, n}] // Max;
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 17 2017 *)

Crossrefs

Cf. A000793, A080742, A080743.

Sequence in context: A355475 A371630 A080742 * A058215 A330542 A166456

Adjacent sequences: A005414 A005415 A005416 * A005418 A005419 A005420

Keyword

nonn,nice,more

Author

N. J. A. Sloane

Extensions

Entry revised by N. J. A. Sloane, Mar 10 2002

Status

approved