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A005417 Maximal period of an n-stage shift register.

%I A005417 M1587 #34 Oct 01 2023 22:38:36
%S A005417 2,6,12,30,60,120,210,420,840,1260,2520,2520,5040,9240,13860,27720,
%T A005417 32760,55440,65520,120120,180180,360360,360360,720720,720720,942480,
%U A005417 1113840
%N A005417 Maximal period of an n-stage shift register.
%C A005417 Maximal order of an element of finite order in GL(2n, Z) or GL(2n+1, Z).
%C A005417 a(n) is the max of the first n numbers in A080742.
%D A005417 H. Lüneburg, Galoisfelder, Kreisteilungskörper und Schieberegisterfolgen. B. I. Wissenschaftsverlag, Mannheim, 1979.
%D A005417 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005417 Benjamin Grossmann, <a href="https://math.stackexchange.com/q/4199260">What is the largest (finite) order of an element of GL(10,Q)?</a>
%H A005417 J. Kuzmanovich and A. Pavlichenkov, <a href="http://www.jstor.org/stable/2695329">Finite groups of matrices whose entries are integers</a>, Amer. Math. Monthly, 109 (2002), 173-186.
%F A005417 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.
%t A005417 (* b,c = a080737 *)
%t A005417 nmax = 26;
%t A005417 kmax = 1200000; (* kmax increased by 100000 until results do not change *)
%t A005417 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]])] ];
%t A005417 orders = Table[{k, b[k]}, {k, 1, kmax}];
%t A005417 c[0] = 2; c[n_] := c[n] = Select[orders, 2n-1 <= #[[2]] <= 2n&][[-1, 1]];
%t A005417 a[n_] := Table[c[m], {m, 0, n}] // Max;
%t A005417 Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Dec 17 2017 *)
%Y A005417 Cf. A000793, A080742, A080743.
%K A005417 nonn,nice,more
%O A005417 0,1
%A A005417 _N. J. A. Sloane_
%E A005417 Entry revised by _N. J. A. Sloane_, Mar 10 2002