Let $H\_n : (\langle 0,1 \rangle \cap \mathbb{Q})^n \to \langle 0,\log\_2 n \rangle, \; H\_n(P) = -\sum\_{i=1}^n P\_i \log\_2 P\_i, \; \sum\_i P\_i = 1$ be the information entropy on rationals. I am looking for an integer-valued analogue $H\_n' : \mathbb{N}^n\to\mathbb{Z}$ satisfying the following: Denote $ K\_x =\sum\_{i=1} C\_{x,i}$ then $H\_n\left(\frac{C\_a}{K\_a}\right) > H\_n\left(\frac{C\_b}{K\_b}\right) $ iff $H\_n'(C\_a) > H\_n'(C\_b).$ The inequality really needs to be strict. In other words, I am looking for a monotonic transform from $H\_n$ to $H\_n'$.


I am unable to figure this out, first I thought that putting $H$ to exponent would suffice but $2^{H\_n(P)} = \prod\_{i=1}^n P\_i^{-P\_i}$ is obviously not integer valued. Then I ran out of ideas. 



Is there an integer-valued analogue of information entropy?
Such a function cannot exist.


Let $I$ be the image of $H\_n$. Then $I$ is a countable dense subset of $[0, \log\_2 n]$. If such a function $H\_n'$ would exist, then $H\_n'(C\_a) = f\big(H\_n\big(\tfrac{C\_a}{K\_a}\big)\big)$, where $f$ is an injective monotonic function $f:I\to\mathbb{Z}$. Such a function $f$ cannot exist: For any $x,y\in I$ there is $z\in I$ with $x < z < y$, but the same is not true in $\mathbb{Z}$.


