PE 0014: Longest Collatz Sequence

The following iterative sequence is defined for the set of positive integers:

• $n \to n/2$ ($n$ is even)
• $n \to 3n + 1$ ($n$ is odd)

Using the rule above and starting with $13$, we generate the following sequence:

$$13\to 40\to 20\to 10\to 5\to 16\to 8\to 4\to 2\to 1.$$

It can be seen that this sequence (starting at $13$ and finishing at $1$) contains $10$ terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at $1$.

Which starting number, under one million, produces the longest chain?

NOTE: Once the chain starts the terms are allowed to go above one million.

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