Digits Cubes Limit

\(f(n)\) gives the sum of the cubed digits of some positive integer \(n.\) For example, \(f(123)=1^3+2^3+3^3=36.\)

If we repeatedly apply this process on each previous result, the following two different behaviors may arise:

• Arrive at a fixed point: For example, beginning with 3, we eventually arrive at 153, and \(f(153)=153.\) We say that 153 is a fixed point.

• Arrive at a limit cycle: For example, beginning with 4, we eventually arrive at 133, and \(f(133) = 55 \implies f(55) = 250 \implies f(250)=133.\) We say that \(\{55,133,250\}\) is a limit cycle.

Let the limit set be the set of all fixed points and limit cycles in the range of \(f(n).\)

Find the sum of all the numbers in the limit set (including the four in pink found above). Note: A coding environment is provided below:

Bonus: Prove that the limit set actually contains finitely many numbers.