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A084828 Maximum number of spheres of radius one that can be packed in a sphere of radius n. 5
1, 2, 13, 32, 68 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=1..5.

Sen Bai, X. Bai, X. Che, X. Wei, Maximal Independent Sets in Heterogeneous Wireless Ad Hoc Networks, IEEE Transactions on Mobile Computing (Volume: 15, Issue: 8, Aug. 1 2016), 2023 - 2033.

Dave Boll, Optimal Packing of Circles and Spheres

Sunil K. Chebolu, Packing Moons Inside the Earth, arXiv:2006.00603 [physics.pop-ph], 2020.

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.

WenQi Huang and Liang Yu, Serial Symmetrical Relocation Algorithm for the Equal Sphere Packing Problem, arXiv preprint arXiv:1202.4149 [cs.DM], 2012. - From N. J. A. Sloane, Jun 14 2012

Hugo Pfoertner, Numerical results for best packing of spheres in sphere.

Hugo Pfoertner, Densest Packing of Spheres in a Sphere. Java visualization.

Yu Liang, Coordinates of sphere centers of 68 spheres of radius 0.20000222, fitting into a container of radius 1. Private communication, Aug 22, 2011


Cf. A121346 Conjectured lower bounds. A084827, A084829, A084825.

Sequence in context: A285096 A177455 A185950 * A100512 A051474 A062708

Adjacent sequences:  A084825 A084826 A084827 * A084829 A084830 A084831




Hugo Pfoertner, Jun 12 2003


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



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Last modified July 31 08:06 EDT 2021. Contains 346369 sequences. (Running on oeis4.)