Longest Collatz sequence

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

\[ \begin{align} & n \rightarrow \frac{n}{2}\ (n\ is\ even)\\ & n \rightarrow 3n + 1\ (n\ is\ odd) \end{align} \]

Using the rule above and starting with \( 13 \), we generate the following sequence:

\[ 13 \rightarrow 40 \rightarrow 20 \rightarrow 10 \rightarrow 5 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1 \]

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.