cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A084828 Maximum number of spheres of radius one that can be packed in a sphere of radius n.

Original entry on oeis.org

1, 2, 13, 32, 68
Offset: 1

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Author

Hugo Pfoertner, Jun 12 2003

Keywords

Comments

a(4) and a(5) are experimental values. Although A121346(5) claims a lower bound of a(5)=68, it is conjectured from extensive numerical search that this value is unachievable and therefore a(5)=67.
The conjecture a(5)=67 was proved wrong by Yu Liang, who found an arrangement of 68 spheres of radius 1 fitting into a sphere of radius 5.
Lower bounds for the next terms are a(6)>=122 and a(7)>=198. See E. Specht's webpage for latest data. - Hugo Pfoertner, Jan 22 2024

Crossrefs

Cf. A121346 (conjectured lower bounds), A084827, A084829, A084825.
Cf. A023393 (2D).

Extensions

Comment and links edited, a(5) from Hugo Pfoertner, Jun 23 2011
a(5) corrected, based on private communication from Yu Liang, by Hugo Pfoertner, Aug 24 2011

A084829 Best packing of m>1 equal spheres in a sphere setting a new density record.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 11, 12, 18, 21, 25, 30, 31, 32, 33, 34, 35, 36, 38, 49, 51, 53, 56, 59, 60, 61
Offset: 1

Views

Author

Hugo Pfoertner, Jun 12 2003

Keywords

Comments

All terms beyond m=9 are only conjectures found by numerical experimentation. The density is defined as the fraction of the volume of the large sphere occupied by the small spheres. For 2 spheres the density is 0.25. The first known configuration with density exceeding 0.5 occurs for 31 spheres.
See the E. Specht link for latest results. - Eduard Baumann, Jan 03 2024

Crossrefs

Extensions

Inserted missing term 30, added comment with conjectured next terms and updated links by Hugo Pfoertner, Jun 24 2011
More terms from Hugo Pfoertner, Aug 25 2013

A342559 Number of equal spheres setting a new density record in relation to the volume of the spherical layer that is occupied by the spheres when arranged touching the surface of a container sphere according to the criterion of maximizing their minimum mutual distance.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 11, 12, 20, 24, 32, 38, 42, 44, 48
Offset: 1

Views

Author

Hugo Pfoertner, Apr 07 2021

Keywords

Comments

This sequence is analogous to A084829, sharing the terms up to 12, but with the restriction that only arrangements are considered in which all small spheres touch the surface of the container sphere and the inner area remains empty. Formal proofs of optimality exist only for arrangements up to and including 14 and for 24 spheres, but the last improvements in the range of the specified terms were found before 1994. For references and links see A080865.
A conjectured continuation after the term 48 is 72, 78, 84, 92, 98, 120.
The linked illustration also shows a fitted curve estimating the minimum density achievable by optimal solutions of the Tammes problem for large n. The fitted equation is rho_min(n) = 0.565854 - 1/(0.566242*n + 2.67822). For comparison, consider the highest attainable density of spheres arranged in a flat hexagonal grid. This density is 0.604599788... = Pi * sqrt(3)/9. Achieving this density is made more difficult in the curved surface layer of a sphere because with large n there must always be 12 neighborhoods where the spheres packed in this layer can only have 5 nearest neighbors.

Examples

			  a(n)  Volume fraction in layer (rounded)
   2    0.25000
   3    0.30000
   4    0.36364
   6    0.42857
   8    0.43853
   9    0.45000
  10    0.45152
  11    0.46397
  12    0.50615
  20    0.51162
  24    0.52941
  32    0.53205
  38    0.53373
  42    0.53439
  44    0.54286
  48    0.54993
		

Crossrefs

Showing 1-3 of 3 results.