Longest Collatz sequence

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

$n → n/2$ ($n$ is even) $n → 3n + 1$ ($n$ is odd)

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

$$ 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1$$

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.