A080742 -id:A080742 - cp's OEIS frontend

A080738 Array read by rows in which 0th row is {1,2}; for n>0, n-th row gives finite orders of 2n X 2n integer matrices that are not orders of 2n-1 X 2n-1 integer matrices.

1, 2, 3, 4, 6, 5, 8, 10, 12, 7, 9, 14, 15, 18, 20, 24, 30, 16, 21, 28, 36, 40, 42, 60, 11, 22, 35, 45, 48, 56, 70, 72, 84, 90, 120, 13, 26, 33, 44, 63, 66, 80, 105, 126, 140, 168, 180, 210, 39, 52, 55, 78, 88, 110, 112, 132, 144, 240, 252, 280, 360, 420, 17, 32, 34, 65, 77

Offset: 0

N. J. A. Sloane, Mar 08 2003

A080739 gives number of elements in n-th row.
If k appears in row n, then k-fold rotational symmetry is compatible with some 2n- (or higher) dimensional crystallographic symmetry. - Andrey Zabolotskiy, Jul 08 2017
The set of finite orders of n X n integer matrices = {m : A080737(m) <= n}. This set is also {a(i) : 1<=i <= Sum_{0<=j<=n/2} A080739(j)}. - Günter Rote, Sep 18 2023

			The array begins:
  1, 2;
  3, 4,  6;
  5, 8, 10, 12;
  7, 9, 14, 15, 18, 20, 24, 30;
  ...

Reinhard Zumkeller, Rows n = 0..25 of triangle, flattened
J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.
W. Steurer and S. Deloudi, Higher-Dimensional Approach. In: Crystallography of Quasicrystals. Springer Series in Materials Science, vol 126. Springer, Berlin, Heidelberg, 2009.

Cf. A080737, A080739, A080740, A080741, A080742.

import Data.Map (singleton, deleteFindMin, insertWith)
a080738 n k = a080738_tabf !! n !! k
a080738_row n = a080738_tabf !! n
a080738_tabf = f 3 (drop 2 a080737_list) 3 (singleton 0 [2,1]) where
   f i xs'@(x:xs) till m
     | i > till  = (reverse row) : f i xs' (3 * head row) m'
     | otherwise = f (i + 1) xs till (insertWith (++) (div x 2) [i] m)
     where ((_,row),m')  = deleteFindMin m
-- Reinhard Zumkeller, Jun 13 2012

a080737[1] = a080737[2] = 0; a080737[p_?PrimeQ] := a080737[p] = p-1; a080737[n_] := a080737[n] = If[ Length[fi = FactorInteger[n]] == 1, EulerPhi[n], Total[ a080737 /@ (fi[[All, 1]]^fi[[All, 2]])]]; orders = Table[{n, a080737[n]}, {n, 1, 420}]; row[0] = {1, 2};row[n_] := Select[ orders, 2n-1 <= #[[2]] <= 2n & ][[All, 1]]; A080738 = Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Jun 20 2012 *)

More terms from Vladeta Jovovic, Mar 09 2003

A080741 Smallest element of n-th row of A080738.

Original entry on oeis.org

1, 3, 5, 7, 16, 11, 13, 39, 17, 19, 25, 23, 69, 115, 29, 31, 64, 155, 37, 111, 41, 43, 129, 47, 141, 235, 53, 81, 265, 59, 61, 183, 128, 67, 201, 71, 73, 219, 365, 79, 237, 83, 249, 415, 89, 267, 445, 623, 97, 291, 101, 103, 309, 107, 109, 121, 113, 339, 565, 791, 1417

Offset: 0

N. J. A. Sloane, Mar 08 2003

J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Cf. A080737, A080738, A080740, A080742.

a080741 n k = a080741_list !! n
a080741_list = map head a080738_tabf  -- Reinhard Zumkeller, Jun 13 2012

More terms from Vladeta Jovovic, Mar 09 2003

A005417 Maximal period of an n-stage shift register.

Original entry on oeis.org

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

N. J. A. Sloane

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.

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).

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.

Cf. A000793, A080742, A080743.

(* 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 *)

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.

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

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A080738 Array read by rows in which 0th row is {1,2}; for n>0, n-th row gives finite orders of 2n X 2n integer matrices that are not orders of 2n-1 X 2n-1 integer matrices.

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A080741 Smallest element of n-th row of A080738.

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A005417 Maximal period of an n-stage shift register.

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