**Original version.**
I believe that it is an elementary question, already discussed somewhere. But I just have no idea of how to start it properly. Take a positive integer $n=n_1$ and compute its sum of digits $n_2=S(n)=S_{10}(n)$ in the decimal system. If the newer number $n_2$ is greater than $10$, then compute the sum $n_3=S(n_2)$ of its digits, and continue this iteration $n_k=S(n_{k-1})$ unless you get a number $n^* =n_\infty$ in the range $1\le n^* \le 9$. Is $n^*$ uniformly distributed in the set $\lbrace 1,2,\dots,9\rbrace$? If this is not true in the decimal systems, what can be said in the other systems?

I just learned yesterday about the Feng shui system of determining what kind of problems/advantages one can get according to the house number, say $n$, of his/her home. This depends on the above $n^* $. I do not seriously count on the conclusions but I am curious whether $n^* $ is sufficiently democratic.

**Edit.** The question was immediately realized as obvious, because $n^*$ is the residue modulo $9$ (with 0 replaced by 9), and this works in any base as well. So the Feng shui function is really trivial, but one can deal with less trivial ones.

Let me fix $m$ and define $Q_m(n)$ as the sum of $m$th powers of decimal digits of a positive integer $n$. What can be said about the sequence of iterations $n_k=Q_m(n_{k-1})$ for a given integer $n_0$? How long can the (minimal) period be for a fixed $m$? And what can be said about the distribution of the purely periodic tails?

I hope that the question is still elementary.