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%I #40 Sep 25 2023 08:54:38
%S 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,
%T 35,45,48,56,70,72,84,90,120,13,26,33,44,63,66,80,105,126,140,168,180,
%U 210,39,52,55,78,88,110,112,132,144,240,252,280,360,420,17,32,34,65,77
%N 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.
%C A080739 gives number of elements in n-th row.
%C 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
%C The set of finite orders of n X n integer matrices = {m : A080737(m) <= n}.
%C This set is also {a(i) : 1<=i <= Sum_{0<=j<=n/2} A080739(j)}. - _Günter Rote_, Sep 18 2023
%H Reinhard Zumkeller, <a href="/A080738/b080738.txt">Rows n = 0..25 of triangle, flattened</a>
%H J. Bamberg, G. Cairns and D. Kilminster, <a href="http://www.jstor.org/stable/3647934">The crystallographic restriction, permutations and Goldbach's conjecture</a>, Amer. Math. Monthly, 110 (March 2003), 202-209.
%H W. Steurer and S. Deloudi, <a href="https://doi.org/10.1007/978-3-642-01899-2_3">Higher-Dimensional Approach</a>. In: Crystallography of Quasicrystals. Springer Series in Materials Science, vol 126. Springer, Berlin, Heidelberg, 2009.
%e The array begins:
%e 1, 2;
%e 3, 4, 6;
%e 5, 8, 10, 12;
%e 7, 9, 14, 15, 18, 20, 24, 30;
%e ...
%t 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 *)
%o (Haskell)
%o import Data.Map (singleton, deleteFindMin, insertWith)
%o a080738 n k = a080738_tabf !! n !! k
%o a080738_row n = a080738_tabf !! n
%o a080738_tabf = f 3 (drop 2 a080737_list) 3 (singleton 0 [2,1]) where
%o f i xs'@(x:xs) till m
%o | i > till = (reverse row) : f i xs' (3 * head row) m'
%o | otherwise = f (i + 1) xs till (insertWith (++) (div x 2) [i] m)
%o where ((_,row),m') = deleteFindMin m
%o -- _Reinhard Zumkeller_, Jun 13 2012
%Y Cf. A080737, A080739, A080740, A080741, A080742.
%K nonn,tabf,easy,look
%O 0,2
%A _N. J. A. Sloane_, Mar 08 2003
%E More terms from _Vladeta Jovovic_, Mar 09 2003
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