If I have a function f(x) which is defined on the rational numbers where x can be represented as $\frac{a}{b}$ where a and b are mutually prime positive integers and b > a. If x can can be represented without a repeating decimal, then f(x) equals 0. Otherwise, f(x) is a positive integer that contains all of the digits in its repeating decimal expansion in the order they appear in the repeating part.

Thus, for example f($\frac{1}{3}$) would be 3, since $\frac{1}{3}=0.\overline{3}$, f($\frac{1}{7}$) would be 142857 since $\frac{1}{7}=0.\overline{142857}$, f($\frac{8}{275}$) would be 90 since $\frac{8}{275}=0.02\overline{90}$ etc.

Is it possible to construct a frequency distribution for the digits 0-9 across the integers that are returned by f(x) where f(x) is non-zero?

I was initially thinking that all digits would occur in this equally frequently since it is an infinitely large set. However, the mapping f(x) is not actually one-to-one, so I am now wondering if is this an oversimplification.

Is it possible to calculate the relative frequencies of for each of the 10 digits in f(x) across its entire domain?

This is not a homework problem for any course that I am aware of, it's just a problem I came up with on my own one day, and I'm not sure how to solve it, or if it even has a calculable solution.


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