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

0,2

Comments

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

Links

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.

Example

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

Mathematica

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 *)

Prog

(Haskell)
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

Crossrefs

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

Sequence in context: A194507 A118316 A197756 * A032447 A224531 A058213

Adjacent sequences: A080735 A080736 A080737 * A080739 A080740 A080741

Keyword

nonn,tabf,easy,look

Author

N. J. A. Sloane, Mar 08 2003

Extensions

More terms from Vladeta Jovovic, Mar 09 2003

Status

approved