Results about numbers that are related to their decimal representation are usually confined to recreational mathematics. There I have seen mainly questions about individual numbers, like finding a number that is the sum of the cubes of its digits. But I have not yet seen a systematic study of such questions.
A result that prompted me to asked this question was the following from Kurt Hensel's "Zahlentheorie". There he shows that the exponent of highest power of a prime $p$ that divides the number $n$ is $$\frac{s_{n-1} - s_n + 1}{p - 1},$$ where $s_n$ is the sum of the digits in the representation of $n$ with base $p$. From this he derives that the highest power of $p$ that divides $n!$ is $$\mu_n = \frac{n - s_n}{p - 1}.$$ This is a different and much simpler formula than the representation $\mu_n = \sum_{k = 0}^\infty \left\lfloor \frac{n}{p^k} \right\rfloor$ by Legendre, which I usually see in the literature.
I would like to know whether there are more results of this type. To be more precise, I define the not-yet-existing field of "Digit Theory" as the study of number-theoretic questions with help of the positional representation of numbers, and the study of the positional representation itself.
So are there already systematic treatments of "Digit Theory" or of parts of it? Are there other interesting results like that one above?
