I've been trying to analyse by myself for recreational purposes what would be a "better" base to use instead of the common decimal one. Part of what should make a base better is to have simpler/smaller fractional representations. In this sense, for all bases from 2 to 100, I computed the periods of the inverses of the first $10^5$ primes and calculated their average, or, equivalently, the average of the order of all theses bases modulo $p$ in the given range, as given in the expression below. $$\frac{1}{10^5}\sum_{i=1 \\ \text{gcd}(p_i,b)=1}^{10^5}\text{ord}_{p_i}(b)$$ The result is shown in the following graph.
There are significant dips at perfect powers, specially at the squares, which means that, overall, the fractional representations of rationals, using these numbers as a base, are much "smaller" than the rest of the numbers on average. Not only that, but the average is not strictly decreasing as the power increases, and even powers have lower averages than odd powers. Could that have something to do with the fact that the order is even about 2/3 of the time? What explains this behavior? There also seem to be some of the form $5n^2$ with average order slightly higher.