A084828 -id:A084828 - cp's OEIS frontend

A084827 Maximum number of spheres of volume one that can be packed in a sphere of volume n.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 6, 6, 6, 7, 8, 8, 9, 9, 10, 10, 12, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 17, 18, 18, 19, 19, 19, 20, 21, 21, 21, 22, 22, 23, 23, 23, 25, 25, 26, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 34, 35, 36, 36, 38, 38, 38, 38, 39, 39, 40, 40, 42, 42, 42, 43, 43, 44

Offset: 1

Hugo Pfoertner, Jun 09 2003

Higher terms of the sequence are only conjectures derived from numerical results. The first 12 arrangements are identical with the solutions of the Tammes problem (see A080865).

			a(10)=2 because a sphere of volume 10 is slightly too small to cover 3 mutually touching spheres of volume 1. a(27)=13 because the arrangement of 12 spheres plus one central sphere needs exactly a sphere with R=3*r to be contained.

Hugo Pfoertner, Table of n, a(n) for n = 1..132
Hugo Pfoertner, Densest packings of n equal spheres in a sphere of radius 1 (Table of largest possible radii)
Hugo Pfoertner, Numerical results for best packing of spheres in sphere.
Eric Weisstein's World of Mathematics, World of Mathematics: Sphere Packing.

Cf. A084828, A084829, A084824, A080865.

More terms from Hugo Pfoertner, May 09 2005

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

Hugo Pfoertner, Jun 12 2003

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

Dave Boll, Optimal Packing of Circles and Spheres.
WenQi Huang and Liang Yu, Serial Symmetrical Relocation Algorithm for the Equal Sphere Packing Problem, arXiv:1202.4149 [cs.DM], 2012.
Eckard Specht, The best known packings of equal spheres in a sphere, July 2023.
Hugo Pfoertner, Numerical results for best packing of spheres in sphere.
Hugo Pfoertner, Densest Packings of n Equal Spheres in a Sphere of Radius 1 Largest Possible Radii.
Eric Weisstein's World of Mathematics, Sphere Packing.
Jianrong Zhou, Shuo Ren, Kun He, Yanli Liu, and Chu-Min Li, An Efficient Solution Space Exploring and Descent Method for Packing Equal Spheres in a Sphere, arXiv:2305.10023 [cs.CG], 2023.

Cf. A084827, A084826, A084644, A084828, A121346.

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

A121346 Conjectured lower bound for the number of spheres of radius 1 that can be packed in a sphere of radius n.

Original entry on oeis.org

2, 11, 31, 68, 124, 205, 316, 460, 642, 866, 1138, 1461, 1839, 2278, 2781, 3354, 4000, 4724, 5531, 6424, 7409, 8490, 9671, 10956, 12351, 13859, 15485, 17234, 19110, 21116, 23259, 25542, 27969, 30546, 33276, 36164, 39215, 42432, 45821, 49385

Offset: 2

Hugo Pfoertner, Jul 22 2006

nonn

The formula was given by David W. Cantrell in a thread "Packing many equal small spheres into a larger sphere" in the newsgroup sci.math on May 29 2006.

Cantrell's formula can be expressed quite accurately using an easy-to-remember rule of thumb: a(n) = n^2*((3/4)*n - 1). To be even more precise, subtract 1%, i.e., multiply by a factor of 0.99. - Hugo Pfoertner, Jun 12 2025

Hugo Pfoertner, Table of n, a(n) for n = 2..10000
Sen Bai, X. Bai, X. Che, and X. Wei, Maximal Independent Sets in Heterogeneous Wireless Ad Hoc Networks, IEEE Transactions on Mobile Computing (Volume: 15, Issue: 8, Aug. 1 2016), pp. 2023-2033.
David W. Cantrell, Packing many equal small spheres into a large sphere, post in newsgroup sci.math, May 29 2006.
WenQi Huang and Liang Yu, A Quasi Physical Method for the Equal Sphere Packing Problem, in 2011 IEEE 10th International Conference on Trust, Security and Privacy in Computing and Communications.

Cf. A084828, A093825.

A121346[n_] := Floor[n^2*(Pi*(n - 2)*Sqrt[2] + 3)/6];
Array[A121346, 50, 2] (* Paolo Xausa, Jun 12 2025 *)

a(n) = floor(K*(1 - 2*d)/d^3 + 1/(2*d^2)), where d=1/n and K = Pi/(3*sqrt(2)) (A093825).

A084825 Maximum number of spheres of diameter one that can be packed in a cube of edge length n.

Original entry on oeis.org

1, 8, 27, 66

Offset: 1

Hugo Pfoertner, Jun 12 2003

From an extrapolation of Dave Boll's numerical results a(4)~=66 and a(5)~=141 are estimated values for the next terms.

However, E. Specht's data suggest a(5)=135. - Hugo Pfoertner, Jul 08 2025

			a(3)=27 because there is no known better arrangement than the 3*3*3 cubic one that would allow packing more than 27 spheres into a cube of edge length 3.

Dave Boll, Optimal Packing of Circles and Spheres
Hugo Pfoertner, Densest Packings of Equal Spheres in a Cube
Hugo Pfoertner, How to pack 66 spheres of diameter 1 in 4 x 4 x 4 cubic box, YouTube video by Talabfahrer, 2016.
Hugo Pfoertner, Observed number N of spheres in best packing known versus edge length x of cube, comparison against empirical lower limit N > 1.325*x^2*(x-1), (2025).
Eckard Specht, The best known packings of equal spheres in a cube (complete up to N = 1000), 2013.

Cf. A084824, A084828, A084617.

a(4) from Hugo Pfoertner, May 21 2011

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A084827 Maximum number of spheres of volume one that can be packed in a sphere of volume n.

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

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A121346 Conjectured lower bound for the number of spheres of radius 1 that can be packed in a sphere of radius n.

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A084825 Maximum number of spheres of diameter one that can be packed in a cube of edge length n.

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