Factorials, double factorials (product of odd numbers) and powers of 2 come in to play in regards to Volumes of n-balls...
And so too Spherical Harmonics...
Hyperspherical volume element
So... when summing volumes for a unit sphere, then e will naturally also be involved.
pi^e/n! = SUM[V_2n]
Insofar as e relates to the prime number distribution [pi (x) ~ x/ln(x)] specifically, and compound growth generally, that there is some manner of relationship, discovered (and I am unaware of it) or undiscovered, seems evident. The precise nature of this relationship, however, is far less clear.
Keep in mind, however, that the number of conjugacy classes in the Symmetric Group S_n is a partition number:
Since we now know, by the work of Ono et al, that partitions of prime numbers evidence fractal-like behavior, we can also logically surmise that the growth sequences of n-dimensional spaces of dimension p and/or p-1 (and/or p+1) will also be found to exhibit fractal-like behaviors. Think of it this way, and then the root system of a lattice such as E8 (241 is prime, and so too 239...) can, in some manner at least, be thought of as if it were a freeze-framed cross-section of a fractal iterating through multi-dimensional space.
And, insofar as all of this is the case, then Periodicity (e.g. The Crystal Restriction Theorem) and Quasi-periodicity (e.g. Penrose Tilings, related to the Golden Ratio) should also make an appearance is some form. (And so too, for that matter, the Shell theorem that you posted, which has everything to do with theoretical physics...)
As for this...
I need to look more closely at what you've been doing before I can answer.
Raphie, I have a direct link form my equation into Apollonian sphere packing. http://oeis.org/A045506
5 + 2^2 = 9
7 + 3^2 = 16
11 + 5^2 = 36
13 + 6^2 = 49
17 + 8^2 = 81
19 + 9^2 = 100
23 + 11^2 = 144
of course this is linked to the fact that (2^(p-1)-1)/p is congruent to 0 (mod 3), for all primes p greater than 3