A080740 -id:A080740 - cp's OEIS frontend

A080737 a(1) = a(2) = 0; for n > 2, the least dimension of a lattice possessing a symmetry of order n.

0, 0, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 6, 8, 16, 6, 18, 6, 8, 10, 22, 6, 20, 12, 18, 8, 28, 6, 30, 16, 12, 16, 10, 8, 36, 18, 14, 8, 40, 8, 42, 12, 10, 22, 46, 10, 42, 20, 18, 14, 52, 18, 14, 10, 20, 28, 58, 8, 60, 30, 12, 32, 16, 12, 66, 18, 24, 10, 70, 10, 72, 36, 22, 20, 16, 14

Offset: 1

N. J. A. Sloane, Mar 08 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.
Savinien Kreczman, Luca Prigioniero, Eric Rowland, and Manon Stipulanti, Magic numbers in periodic sequences, Univ. Liège (Belgium, 2023). See p. 7.

Cf. A080736, A080738, A080739, A080740, A067240, A000010, A141809.

See A152455 for another version.

a080737 n = a080737_list !! (n-1)
a080737_list = 0 : (map f [2..]) where
f n | mod n 4 == 2 = a080737 $ div n 2
| otherwise = a067240 n
-- Reinhard Zumkeller, Jun 13 2012, Jun 11 2012

a[1] = a[2] = 0; a[p_?PrimeQ] := a[p] = p-1; a[n_] := a[n] = If[Length[fi = FactorInteger[n]] == 1, EulerPhi[n], Total[a /@ (fi[[All, 1]]^fi[[All, 2]])]]; Table[a[n], {n, 1, 78}] (* Jean-François Alcover, Jun 20 2012 *)

for(n=1,78,k=0; if(n>1,f=factor(n); k=sum(j=1,matsize(f)[1],eulerphi(f[j,1]^f[j,2])); if(f[1,1]==2&&f[1,2]==1,k--)); print1(k,",")) \\ Klaus Brockhaus, Mar 10 2003

For n > 2, a(2^r) = 2^(r-1) with r>1, a(p^r) = phi(p^r) with p > 2 prime, r >= 1, where phi is Euler's function A000010; in general if a(Product p_i^e_i) = Sum a(p_i^e_i).

More terms from Klaus Brockhaus, Mar 10 2003

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.

Original entry on oeis.org

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

N. J. A. Sloane, Mar 08 2003

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

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

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.

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

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

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 *)

More terms from Vladeta Jovovic, Mar 09 2003

A080739 Number of elements in n-th row of A080738.

Original entry on oeis.org

2, 3, 4, 8, 7, 11, 13, 14, 20, 26, 30, 42, 43, 49, 64, 65, 82, 95, 107, 129, 143, 156, 186, 203, 235, 267, 295, 334, 372, 415, 466, 514, 581, 637, 707, 783, 859, 960, 1055, 1152, 1281, 1390, 1522, 1679, 1824, 2011, 2190, 2377, 2595, 2816, 3059, 3342, 3616, 3940

Offset: 0

N. J. A. Sloane, Mar 08 2003

J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Cf. A080737, A080738, A080740.

a080739 n k = a080739_list !! n
a080739_list = map length a080738_tabf  -- Reinhard Zumkeller, Jun 13 2012

More terms from Devin Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003

A080742 Largest element of n-th row of A080738.

Original entry on oeis.org

2, 6, 12, 30, 60, 120, 210, 420, 840, 1260, 2520, 2310, 5040, 9240, 13860, 27720, 32760, 55440, 65520, 120120, 180180, 360360, 235620, 720720, 556920, 942480, 1113840, 1531530

Offset: 0

N. J. A. Sloane, Mar 08 2003

J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Cf. A080737, A080738, A080740, A080741, A005417.

a080742 n k = a080742_list !! n
a080742_list = map last a080738_tabf  -- Reinhard Zumkeller, Jun 13 2012

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, 2*10^6}]; row[0] = {1, 2}; row[n_] := Select[orders, 2*n - 1 <= #[[2]] <= 2*n &][[All, 1]]; Table[ row[n] // Last, {n, 0, 27}] (* Jean-François Alcover, Jul 24 2013 *)

Corrected and extended by Vladeta Jovovic, Mar 09 2003

A080741 Smallest element of n-th row of A080738.

Original entry on oeis.org

1, 3, 5, 7, 16, 11, 13, 39, 17, 19, 25, 23, 69, 115, 29, 31, 64, 155, 37, 111, 41, 43, 129, 47, 141, 235, 53, 81, 265, 59, 61, 183, 128, 67, 201, 71, 73, 219, 365, 79, 237, 83, 249, 415, 89, 267, 445, 623, 97, 291, 101, 103, 309, 107, 109, 121, 113, 339, 565, 791, 1417

Offset: 0

N. J. A. Sloane, Mar 08 2003

J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Cf. A080737, A080738, A080740, A080742.

a080741 n k = a080741_list !! n
a080741_list = map head a080738_tabf  -- Reinhard Zumkeller, Jun 13 2012

More terms from Vladeta Jovovic, Mar 09 2003

cp's OEIS Frontend

A080737 a(1) = a(2) = 0; for n > 2, the least dimension of a lattice possessing a symmetry of order n.

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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.

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A080739 Number of elements in n-th row of A080738.

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A080742 Largest element of n-th row of A080738.

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A080741 Smallest element of n-th row of A080738.

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