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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. 8
 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 (list; graph; refs; listen; history; text; internal format)
 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 compartible with some 2n- (or higher) dimensional crystallographic symmetry. - Andrey Zabolotskiy, Jul 08 2017 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, S. Deloudi, Higher-Dimensional Approach. In: Crystallography of Quasicrystals. Springer Series in Materials Science, vol 126. Springer, Berlin, Heidelberg, 2009. FORMULA Set of finite orders of n X n integer matrices = {m : A080737(m) <= n}. EXAMPLE 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,changed AUTHOR N. J. A. Sloane, Mar 08 2003 EXTENSIONS More terms from Vladeta Jovovic, Mar 09 2003 STATUS approved

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