\begin{definition}[Definition:Recurring Digital Invariant]
Let $k \in \Z_{>0}$ be a positive integer.
Let $f: \Z_{>0} \to \Z_{>0}$ be the mapping defined as:
:$\forall m \in \Z_{>0}: \map f m = $ the sum of the $k$th powers of the digits of $n$.
Let $n_0 \in \Z_{>0}$ be a positive integer.
Consider the sequence:
:$s_n = \begin{cases} n_0 & : n = 0 \\
\map f {s_{n - 1} } & : n > 0 \end{cases}$
If:
:$\exists r \in \N_{>0}: s_r = n_0$
then the smallest of the terms $n_0, n_1, \ldots, n_r$ is a '''recurring digital invariant of order $k$'''.
\end{definition}