Definition:Recurring Digital Invariant

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Definition

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 $k$.


Examples

2178

$2178$ is a $4$th order recurring digital invariant:

\(\ds 2178: \ \ \) \(\ds 2^4 + 1^4 + 7^4 + 8^4\) \(=\) \(\ds 16 + 1 + 2401 + 4096\) \(\ds = 6514\)
\(\ds 6514: \ \ \) \(\ds 6^4 + 5^4 + 1^4 + 4^4\) \(=\) \(\ds 1296 + 625 + 1 + 256\) \(\ds = 2178\)

$\blacksquare$


Also see


Sources