Let $x$ be a base $k$ number with digits: $d_1 d_2 \cdots d_n$
(This implies the value of $x$ is $ d_1k^{n-1} +d_2k^{n-2} + \dots +d_{n-1}k+ d_n$)
We define $R(x)$ to be the "digit sum root":
If $x < k$, $R(x) = x$
Otherwise $R(x) = R(d_1 + d_2 + \dots + d_n)$
That is, we keep adding up the digits until we have a single digit. (all in base k)
What interesting properties does the function $R(x)$ have? For example, what can we say about $R(x + y)$? or $R(xy)$? and so on? How can we calculate $R$ efficiently for large $x$?