A080738 -id:A080738 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A080740 a(n) = number of m such that A080737(m) <= 2n.

Original entry on oeis.org

2, 5, 9, 17, 24, 35, 48, 62, 82, 108, 138, 180, 223, 272, 336, 401, 483, 578, 685, 814, 957, 1113, 1299, 1502, 1737, 2004, 2299, 2633, 3005, 3420, 3886, 4400, 4981, 5618, 6325, 7108, 7967, 8927, 9982, 11134, 12415, 13805, 15327, 17006, 18830, 20841

Offset: 0

N. J. A. Sloane, Mar 08 2003

Total number of elements in first n rows of A080738.

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

Partial sums of A080739.

a080740 n k = a080740_list !! n
a080740_list = scanl1 (+) a080739_list  -- Reinhard Zumkeller, Jun 13 2012

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

A004031 Number of n-dimensional crystal systems.

Original entry on oeis.org

1, 1, 4, 7, 33, 59, 251

Offset: 0

N. J. A. Sloane

From Andrey Zabolotskiy, Jul 12 2017: (Start)
From Souvignier (2003): "the unions of all geometric classes intersecting the same set of Bravais flocks is defined to be a crystal system or point-group system. <...> This means that two geometric classes belong to the same crystal system if for any representative of the first class there is a representative of the other class such that the representatives have GL(n,Q)-conjugate Bravais groups. <...> The definition for crystal systems as given by Brown et al. (1978) therefore is only valid in dimensions up to 4, where it coincides with the more general definition adopted here."
For dimension 6, Souvignier (2003) uses old incorrect CARAT data, but the error affected only geometric classes and finer classification, so the data for crystal systems must be correct.
Among 33 4-dimensional crystal systems, 7 are enantiomorphic.
Coincides with the number of n-dimensional Bravais systems for n<5 (only).
(End)

P. Engel, "Geometric crystallography," in P. M. Gruber and J. M. Wills, editors, Handbook of Convex Geometry. North-Holland, Amsterdam, Vol. B, pp. 989-1041.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978, p. 52. Corrections.
J. Neubüser, W. Plesken, and H. Wondratschek, An emendatory discussion on defining crystal systems, Commun. Math. Chem., 10 (1981), 77-96.
W. Plesken and T. Schulz, CARAT Homepage
W. Plesken and T. Schulz, CARAT Homepage [Cached copy in pdf format (without subsidiary pages), with permission]
B. Souvignier, Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6, Acta Cryst., A59 (2003), 210-220.

Cf. A004026-A004029, A004032, A006226, A006227, A080738.

a(5)-a(6) from Souvignier (2003) by Andrey Zabolotskiy, Jul 12 2017

A180042 The possible orders of cyclic groups that can be realized as holonomy groups of crystallographic groups in dimension 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, 30

Offset: 1

Jonathan Vos Post, Jan 14 2011

In Lutowski's article the CARAT system is used to calculate a list of all isomorphism classes of 7-dimensional Bieberbach groups with cyclic holonomy group. The final list of 316 groups is presented on the undated link by the same author.

Sorted union of first floor(7/2)+1 = 4 rows of A080738. - Andrey Zabolotskiy, Jul 10 2017

H. Hiller, The Crystallographic Restriction in Higher Dimensions, Acta Cryst. (1985), A41, 541-544.
Rafal Lutowski, Seven dimensional flat manifolds with a cyclic holonomy group.
R. Lutowski, A list of 7-dimensional Bieberbach groups with cyclic holonomy, Jan 13, 2011.
W. Plesken and T. Schulz, The CARAT Homepage
W. Plesken and T. Schulz, CARAT Homepage [Cached copy in pdf format (without subsidiary pages), with permission]
W. Plesken and T. Schulz, Introduction to CARAT [Cached copy in pdf format (without subsidiary pages), with permission]

cp's OEIS Frontend

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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Haskell

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A080741 Smallest element of n-th row of A080738.

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Haskell

Extensions

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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A080740 a(n) = number of m such that A080737(m) <= 2n.

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A004031 Number of n-dimensional crystal systems.

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A180042 The possible orders of cyclic groups that can be realized as holonomy groups of crystallographic groups in dimension 7.

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