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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.

%I A080738 #44 Jun 05 2024 01:20:56
%S A080738 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 A080738 35,45,48,56,70,72,84,90,120,13,26,33,44,63,66,80,105,126,140,168,180,
%U A080738 210,39,52,55,78,88,110,112,132,144,240,252,280,360,420,17,32,34,65,77
%N 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.
%C A080738 A080739 gives number of elements in n-th row.
%C A080738 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 A080738 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
%H A080738 Reinhard Zumkeller, <a href="/A080738/b080738.txt">Rows n = 0..25 of triangle, flattened</a>
%H A080738 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 A080738 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 A080738 The array begins:
%e A080738   1, 2;
%e A080738   3, 4,  6;
%e A080738   5, 8, 10, 12;
%e A080738   7, 9, 14, 15, 18, 20, 24, 30;
%e A080738   ...
%t A080738 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 A080738 (Haskell)
%o A080738 import Data.Map (singleton, deleteFindMin, insertWith)
%o A080738 a080738 n k = a080738_tabf !! n !! k
%o A080738 a080738_row n = a080738_tabf !! n
%o A080738 a080738_tabf = f 3 (drop 2 a080737_list) 3 (singleton 0 [2,1]) where
%o A080738    f i xs'@(x:xs) till m
%o A080738      | i > till  = (reverse row) : f i xs' (3 * head row) m'
%o A080738      | otherwise = f (i + 1) xs till (insertWith (++) (div x 2) [i] m)
%o A080738      where ((_,row),m')  = deleteFindMin m
%o A080738 -- _Reinhard Zumkeller_, Jun 13 2012
%Y A080738 Cf. A080737, A080739, A080740, A080741, A080742.
%K A080738 nonn,tabf,easy,look
%O A080738 0,2
%A A080738 _N. J. A. Sloane_, Mar 08 2003
%E A080738 More terms from _Vladeta Jovovic_, Mar 09 2003