N.b. the article uses d for the dimension, and n for the moment index. I haven't cranked out the details, but I believe that this is already true just assuming the fourth moments of the distance are integers. That is, for n != 1, 2, or 4, the fourth moment is never an integer. Idea of the "brute force" proof:

Take the formula in the article for the 4th moment of the d-dimensional sphere, which is always a rational number. Basically, for n=2^k, the denominator should be divisible by a larger power of two than the numerator (specifically, if I crunched the numbers right, the gap should be k-2). When n is not a power of two, then for any odd prime p dividing n, I believe the denominator should be divisible by a larger power of p than the numerator. This requires calculating exactly how many powers of p divide various factorial expressions, but you get the idea.