A080738 - OEIS

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

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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