_{This problem is a programming version of Problem 95 from projecteuler.net}

The proper divisors of a number are all the divisors excluding the number itself. For example, the proper divisors of are and . As the sum of these divisors is equal to , we call it a perfect number.

Interestingly the sum of the proper divisors of is and the sum of the proper divisors of is , forming a chain of two numbers. For this reason, and are called an amicable pair.

Perhaps less well known are longer chains. For example, starting with , we form a chain of five numbers:

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 .