There are many "dimensionless" ratios: choosing $R$ as the linear measurement is arbitrary.
For instance, the ratio of the volume of the sphere to the volume of the circumscribed cube
has a maximum at $n=1$.  The ratio to the volume of the inscribed cube never attains a maximum.  There are intermediate geometrically-related "midscribed" cubes, where all faces of some dimension are tangent to the unit sphere.  Here is the graph for the ratio of
volumes when the codimension 2 faces of a cube are tangent. It attains the maximum for $n = 12$ (just barely more than for $n=11$).  There are many other reasonable dimensionless comparisons, for instance comparing to a simplex, etc. etc.  Since the Gamma function
grows super-exponentially, these simple geometric variations tend to shift the maximum --- there's nothing special about 5 or 7. 

&nbsp;&nbsp;&nbsp;&nbsp;![alt text][1x] &nbsp;&nbsp;[<sup>(source: Wayback Machine)</sup>][1]

  [1]: https://web.archive.org/web/20130219082438im_/http://dl.dropbox.com/u/5390048/MidScribed.jpg
  [1x]: https://i.sstatic.net/On6Rd.jpg