Definition:Cubic Recurring Digital Invariant

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Definition

A cubic recurring digital invariant is a recurring digital invariant of order $3$.


Sequence

The sequence of cubic recurring digital invariants is:

$55, 136, 160, 919$


Examples

55

$55$ is a cubic recurring digital invariant:

\(\ds 55: \ \ \) \(\ds 5^3 + 5^3\) \(=\) \(\ds 125 + 125\) \(\ds = 250\)
\(\ds 250: \ \ \) \(\ds 2^3 + 5^3 + 0^3\) \(=\) \(\ds 8 + 125 + 0\) \(\ds = 133\)
\(\ds 133: \ \ \) \(\ds 1^3 + 3^3 + 3^3\) \(=\) \(\ds 1 + 27 + 27\) \(\ds = 55\)

$\blacksquare$


136

$136$ is a cubic recurring digital invariant:

\(\ds 136: \ \ \) \(\ds 1^3 + 3^3 + 6^3\) \(=\) \(\ds 1 + 27 + 216\) \(\ds = 244\)
\(\ds 244: \ \ \) \(\ds 2^3 + 4^3 + 4^3\) \(=\) \(\ds 8 + 64 + 64\) \(\ds = 136\)

$\blacksquare$


160

$160$ is a cubic recurring digital invariant:

\(\ds 160: \ \ \) \(\ds 1^3 + 6^3 + 0^3\) \(=\) \(\ds 1 + 216 + 0\) \(\ds = 217\)
\(\ds 217: \ \ \) \(\ds 2^3 + 1^3 + 7^3\) \(=\) \(\ds 8 + 1 + 343\) \(\ds = 352\)
\(\ds 352: \ \ \) \(\ds 3^3 + 5^3 + 2^3\) \(=\) \(\ds 27 + 125 + 8\) \(\ds = 160\)

$\blacksquare$


919

$919$ is a cubic recurring digital invariant:

\(\ds 919: \ \ \) \(\ds 9^3 + 1^3 + 9^3\) \(=\) \(\ds 729 + 1 + 729\) \(\ds = 1459\)
\(\ds 1459: \ \ \) \(\ds 1^3 + 4^3 + 5^3 + 9^3\) \(=\) \(\ds 1 + 64 + 125 + 729\) \(\ds = 919\)

$\blacksquare$


Sources