Amicable Chains
The proper divisors of a number are all the divisors excluding the number itself. For example, the proper divisors of $28$ are $1$, $2$, $4$, $7$, and $14$. As the sum of these divisors is equal to $28$, we call it a perfect number.
Interestingly the sum of the proper divisors of $220$ is $284$ and the sum of the proper divisors of $284$ is $220$, forming a chain of two numbers. For this reason, $220$ and $284$ are called an amicable pair.
Perhaps less well known are longer chains. For example, starting with $12496$, we form a chain of five numbers: $$12496 \to 14288 \to 15472 \to 14536 \to 14264 (\to 12496 \to \cdots)$$
Since this chain returns to its starting point, it is called an amicable chain.
Find the smallest member of the longest amicable chain with no element exceeding one million.