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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $55$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $55$
- Weisstein, Eric W. "Recurring Digital Invariant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RecurringDigitalInvariant.html