A004026 -id:A004026 - cp's OEIS frontend

A037075 Number of eutactic lattices in dimension n.

Original entry on oeis.org

1, 2, 5, 16, 118

Offset: 1

J. Martinet (martinet(AT)math.u-bordeaux.fr), N. J. A. Sloane

J. Martinet, Les réseaux parfaits des espaces Euclidiens, Masson, Paris, 1996.
J. Martinet, Perfect Lattices in Euclidean Spaces, Springer-Verlag, NY, 2003.
G. Nebe, Review of J. Martinet, Perfect Lattices in Euclidean Spaces, Bull. Amer. Math. Soc., 41 (No. 4, 2004), 529-533.

J. Martinet and B. Venkov, Les réseaux fortement eutactiques, pp. 112-132 in Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, ed. J. Martinet, L'Enseignement Mathématique, Geneva, 2001.
B. Venkov, Réseaux et designs sphériques, pp. 10-86 in Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, ed. J. Martinet, L'Enseignement Mathématique, Geneva, 2001.

Cf. A065536, A065535, A004026.

A065536 Number of strongly eutactic lattices in dimension n.

Original entry on oeis.org

1, 2, 3, 6, 9, 21

Offset: 1

N. J. A. Sloane, Nov 16 2001

J. Martinet, Les reseaux parfaits des espaces Euclidiens, Masson, Paris, 1996.
J. Martinet, Perfect Lattices in Euclidean Spaces, Springer-Verlag, NY, 2003.
G. Nebe, Review of J. Martinet, Perfect Lattices in Euclidean Spaces, Bull. Amer. Math. Soc., 41 (No. 4, 2004), 529-533.

Christian Batut and Jacques Martinet, Lattices and spherical designs, 2005-2014. See pages 12-13.
J. Martinet and B. Venkov, Les réseaux fortement eutactiques, pp. 112-132 in Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, ed. J. Martinet, L'Enseignement Mathématique, Geneva, 2001.

Cf. A037075, A065535, A004026.

It is known that a(6) >= 19, a(7) >= 10.

a(5) corrected and a(6) added from Batut & Martinet by Andrey Zabolotskiy, Feb 20 2021

A065535 Number of strongly perfect lattices in dimension n.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0

Offset: 1

N. J. A. Sloane, Nov 16 2001

It is known that a(12) through a(24) are at least 1, 0, 1, 0, 3, 0, 1, 0, 1, 1, 5, 4, 2 respectively.

In this sequence, the dual pairs of lattices are counted as one if they are both strongly perfect (it is not always so). E.g., in dimensions 6, 7, 10 there are two strongly perfect lattices, forming a dual pair, but in dimension 21 there is a strongly perfect lattice which has a not strongly perfect dual. - Andrey Zabolotskiy, Feb 20 2021

J. Martinet, Les réseaux parfaits des espaces euclidiens, Masson, Paris, 1996.
J. Martinet, Perfect Lattices in Euclidean Spaces, Springer-Verlag, NY, 2003. See Section 16.2.

B. Venkov, Réseaux et designs sphériques, pp. 10-86 in Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, ed. J. Martinet, L'Enseignement Mathématique, Geneva, 2001.
Jacques Martinet, Known strongly perfect lattices, 2002-2020.

Cf. A037075, A065536, A004026.

A122079 Kissing numbers (divided by 2) of 8-dimensional perfect lattices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 63, 64, 65, 66, 67, 68, 69, 70, 71, 75, 120

Offset: 1

N. J. A. Sloane, Oct 18 2006

M. Dutour Sikiric, A. Schuermann and F. Vallentin, Classification of eight-dimensional perfect forms, Preprint, 2006.

Cf. A004026, A122080.

A122080 Kissing numbers of 8-dimensional perfect lattices.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 120, 126, 128, 130, 132, 134, 136, 138, 140, 142, 150, 240

Offset: 1

N. J. A. Sloane, Oct 18 2006

Mathieu Dutour Sikirić, Achill Schürmann, and Frank Vallentin, Classification of eight-dimensional perfect forms, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 21-32; arXiv preprint, arXiv:math/0609388 [math.NT], 2006-2009.

Cf. A004026, A122079.

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

A033689 Number of extreme quadratic forms or lattices in dimension n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 30, 2408

Offset: 1

N. J. A. Sloane

A lattice is extreme if and only if it is perfect and eutactic. - Andrey Zabolotskiy, Feb 20 2021

J. H. Conway and N. J. A. Sloane, Low-dimensional lattices III: perfect forms, Proc. Royal Soc. London, A 418 (1988), 43-80.
M. Dutour Sikiric, A. Schuermann and F. Vallentin, Classification of eight-dimensional perfect forms, Preprint, 2006.
P. M. Gruber, Convex and Discrete Geometry, Springer, 2007; p. 439
D.-O. Jaquet, Classification des réseaux dans R^7 (via la notion de formes parfaites), Journées Arithmétiques, 1989 (Luminy, 1989). Asterisque No. 198-200 (1991), 7-8, 177-185 (1992).
J. Martinet, Les réseaux parfaits des espaces Euclidiens, Masson, Paris, 1996, p. 175.
J. Martinet, Perfect Lattices in Euclidean Spaces, Springer-Verlag, NY, 2003.
G. Nebe, Review of J. Martinet, Perfect Lattices in Euclidean Spaces, Bull. Amer. Math. Soc., 41 (No. 4, 2004), 529-533.
A. Schuermann, Enumerating perfect forms, Contemporary Math., 493 (2009), 359-377. [From N. J. A. Sloane, Jan 21 2010]

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 3rd edition, 1999, see Preface to 3rd Ed., especially the page that was omitted by the publisher between pages xx and xxi!
D.-O. Jaquet and F. Sigrist, Formes quadratiques contigües à D_7, C. R. Acad. Sci. Paris Ser. I Math. 309 (1989), no. 10, 641-644.
J. Martinet and B. Venkov, Les réseaux fortement eutactiques, pp. 112-132 in Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, ed. J. Martinet, L'Enseignement Mathématique, Geneva, 2001.
C. Riener, On extreme forms in dimension 8, J. Théor. Nombres Bordeaux 18 (2006), no. 3, 677-682.
B. Venkov, Réseaux et designs sphériques, pp. 10-86 in Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, ed. J. Martinet, L'Enseignement Mathématique, Geneva, 2001.

Cf. A004026 (perfect), A037075 (eutactic).

a(8) = 2408 was calculated by G. Nebe's student Cordian Riener - communicated by G. Nebe, Oct 11 2005. He found this number by checking the complete list of 10916 perfect lattices in 8 dimensions (see A004026).

cp's OEIS Frontend

A037075 Number of eutactic lattices in dimension n.

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A065536 Number of strongly eutactic lattices in dimension n.

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A065535 Number of strongly perfect lattices in dimension n.

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A122079 Kissing numbers (divided by 2) of 8-dimensional perfect lattices.

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A122080 Kissing numbers of 8-dimensional perfect lattices.

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A004031 Number of n-dimensional crystal systems.

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A033689 Number of extreme quadratic forms or lattices in dimension n.

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