
COMMENTS

Two additional dimensions have been solved: a(8) = 240, a(24) = 196560 [Odlyzko and Sloane; Levenstein].
Lower bounds for a(5) onwards are 40, 72, 126, 240 (exact), 306, 500, ...  N. J. A. Sloane, May 15 2015
It seems that, while n is even, a lower bound for a(n) can be written as f(n) = (2n + 2^n)  k(n)^2, where k(2) = 2, for n > 2, k(n) = 2^(n/2)  q, q = {2^t, 3*2^t}, t is integer, t > 0, 2 <= q <= n, and f(n) <= a(n) (Note: for n <= 24, q = n at n = {4, 6, 8, 24}, q = 0 at n = 2).  Sergey Pavlov, Mar 17 2017
It also seems that, while n is even, an upper bound for a(n) can be written as f(n) = (2n + 2^n)  k(n)^2, where k(2) = 0, for n > 2, k(n) = 2^(n/2)  q, q = {2^t, 3*2^t}, t is integer, t > 0, n <= q <= 2n, and f(n) >= a(n) (Note: for n <= 24, q = n at n = {2, 4, 12, 16}).  Sergey Pavlov, Mar 19 2017


REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", SpringerVerlag, 3rd. ed., Chap. 3, esp. pp. xxi, 23, etc.
Musin, Oleg Rustumovich. "The problem of the twentyfive spheres." Russian Mathematical Surveys 58.4 (2003): 794795.


LINKS

Table of n, a(n) for n=1..4.
Eiichi Bannai and N. J. A. Sloane, Uniqueness of certain spherical codes, Canad. J. Math. 33 (1981), no. 2, 437449.
J. Leech, The problem of the thirteen spheres, Math. Gaz., 40 (1956), 2223.
V. I. Levenshtein, On bounds for packings in ndimensional Euclidean space, Dokl. Akad. Nauk., 245 (1979), 12991303; English translation in Soviet Math. Doklady, 20 (1979), 417421.
Mittelmann, Hans D.; Vallentin, Frank, "High accuracy semidefinite programming bounds for kissing numbers." Exp. Math. (2009), no. 19, 174178.
G. Nebe and N. J. A. Sloane, Table of highest kissing numbers known
A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in n dimensions, J. Combin. Theory Ser. A 26 (1979), no. 2, 210214.
Wikipedia, Kissing number problem
