A321015 -id:A321015 - cp's OEIS frontend

A321014 Number of divisors of n which are greater than 3.

0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 5, 1, 4, 2, 2, 3, 6, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 7, 2, 4, 2, 4, 1, 5, 3, 6, 2, 2, 1, 9, 1, 2, 4, 5, 3, 5, 1, 4, 2, 6, 1, 9, 1, 2, 4, 4, 3, 5, 1, 8, 3, 2, 1, 9, 3, 2, 2, 6, 1, 9, 3, 4, 2, 2

Offset: 1

N. J. A. Sloane, Nov 04 2018

Marjorie Senechal, "Introduction to lattice geometry." In M. Waldschmidt et al., eds., From Number Theory to Physics, pp. 476-495. Springer, Berlin, Heidelberg, 1992. See Cor. 3.7.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A001620, A321015.
A072527 is a shifted version.
Column k=4 of A135539.

d2:=proc(n) local c;
if n <= 3 then return(0); fi;
c:=NumberTheory[tau](n)-1;
if (n mod 2)=0 then c:=c-1; fi;
if (n mod 3)=0 then c:=c-1; fi; c; end;
[seq(d2(n),n=1..120)];

nmax = 94; Rest[CoefficientList[Series[Sum[x^k/(1 - x^k), {k, 4, nmax}], {x, 0, nmax}], x]] (* Ilya Gutkovskiy, Nov 07 2018 *)

a(n) = sumdiv(n, d, d>3); \\ Michel Marcus, Nov 06 2018

a(n) = numdiv(n) - 3 + !!(n%2) + !!(n%3) \\ David A. Corneth, Nov 07 2018

my(N=100, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=4} x^k/(1 - x^k). - Ilya Gutkovskiy, Nov 06 2018
a(n) = Sum_{d|n, d>3} 1. - Wesley Ivan Hurt, Apr 28 2020
G.f.: Sum_{k>=1} x^(4*k)/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 17/6), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

A071880 Number of combinatorial types of n-dimensional parallelohedra.

Original entry on oeis.org

1, 1, 2, 5, 52, 103769

Offset: 0

N. J. A. Sloane, Jun 10 2002

a(n) is the number of topologically distinct shapes the Voronoi cell (or Vocell) of an n-dimensional lattice can have.
a(n) is the number of combinatorially distinct parallelotopes that tile R^n. Dirichlet proved a(2) = 2, Fedorov showed a(3) = 5, while a(4) = 52 is due to Delone as corrected by Stogrin, and a(5) = 103769 to Engel. - Jonathan Sondow, May 26 2017
The papers by Dutuor Sikiric, Garber et al say that actually a(5) = 110244. The claim that every parallelotope is a Voronoi cell of some lattice in R^n up to an affine transformation is a conjecture open for n > 5. - Andrey Zabolotskiy, Feb 20 2021

			In 1 dimension: the Vocell is an interval (1 possible shape)
In 2 dimensions: a hexagon or rectangle (2 possible shapes)
In 3 dimensions: truncated octahedron, hexarhombic dodecahedron, rhombic dodecahedron, hexagonal prism, cuboid (5 possible shapes)

J. H. Conway, The Sensual Quadratic Form.
E. S. Fedorov, An Introduction to the Theory of Figures. Notices of the Imperial Petersburg Mineralogical Society, 2nd series, vol. 21, 1-279, 1885. (English translation in Symmetry of crystals, ACA Monograph no. 7, 50-131, 1971.)

David Austin, Fedorov's Five Parallelohedra, AMS Feature Column, 2017.
B. N. Delaunay, Sur la partition régulière de l'espace à 4-dimensions. Première partie, Izv. Akad. Nauk SSSR Otdel. Fiz.-Mat. Nauk, 79-110, 1929.
B. N. Delaunay, Sur la partition régulière de l'espace à 4-dimensions. Deuxième partie, Izv. Akad. Nauk SSSR Otdel. Fiz.-Mat. Nauk, 145-164, 1929.
Lejeune G. Dirichlet, Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen, J. reine angew. Math., 40 209-227 (1850); [Oeuvre Vl. II, p. 41-59].
Mathieu Dutour Sikirić, Alexey Garber, Achill Schürmann, Clara Waldmann, The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices, Acta Crystallographica A72 (2016), 673-683; arXiv:1507.00238 [math.MG], 2015-2016.
Mathieu Dutour Sikirić, Alexey Garber, and Alexander Magazinov, On the Voronoi Conjecture for Combinatorially Voronoi Parallelohedra in Dimension 5, SIAM J. Discrete Math., 34(4), 2481-2501 (2020).
P. Engel, The contraction types of parallelohedra in E^5, Acta Cryst. A 56 (2000), 491-496.
Alexey Garber and Alexander Magazinov, Voronoi conjecture for five-dimensional parallelohedra, arXiv:1906.05193 [math.CO], 2019-2020.
M. I. Stogrin, Regular Dirichlet-Voronoi partitions for the second triclinic group, Trudy Matematicheskogo Instituta imeni V. A. Steklova, 123 (1973) [in Russian] = Proceedings of the Steklov Institute of Mathematics, 123 (1973).
Wikipedia, Parallelohedron

Cf. A071881, A071882, A321015.

Corrected by J. H. Conway, Dec 25 2003

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A321014 Number of divisors of n which are greater than 3.

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A071880 Number of combinatorial types of n-dimensional parallelohedra.

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