Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space.
The values are proved for the cases n = 1, 2, 3, 8, 24, but only conjectural for n = 4, 5, 6, 7, 12, 16.
Reference: J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See pp. xix and 15.
The A-numbers refer to the decimal expansion of the corresponding constant in the OEIS.
N. J. A. Sloane, Feb. 10 2013, and Andrey Zabolotskiy, Dec. 30 2023

dim.      center-density            A-number           density                   A-number
      exact          decimal                 exact                 decimal
1     1/2            0.5000000000   -        1                     1             -
2     1/sqrt(12)     0.2886751347   A020769  (1/6)*3^(1/2)*Pi      0.9068996827  A093766
3     1/sqrt(32)     0.1767766952   A020789  (1/6)*2^(1/2)*Pi      0.7404804898  A093825
4     1/8            0.1250000000   -        (1/16)*Pi^2           0.6168502752  A222068
5     1/(8*sqrt(2))  0.08838834762  A222066  (1/30)*2^(1/2)*Pi^2   0.4652576133  A222069
6     1/(8*sqrt(3))  0.07216878367  A222067  (1/144)*3^(1/2)*Pi^3  0.3729475457  A222070
7     1/16           0.06250000000  -        (1/105)*Pi^3          0.2952978732  A222071
8     1/16           0.06250000000  -        (1/384)*Pi^4          0.2536695081  A222072
12    1/27           0.03703703704  A021031  (1/19440)*Pi^6        0.0494541766  -
16    1/16           0.06250000000  -        (1/645120)*Pi^8       0.0147081644  -
24    1              1              -        (1/479001600)*Pi^12   0.0019295743  A260646