Problem 95

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:

$$ 12496 \rightarrow 14288 \rightarrow 15472 \rightarrow 14536 \rightarrow 14264 (\rightarrow 12496 \rightarrow \ldots)$$

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.

亲和数链

一个数除了其本身之外的因数称为真因数。例如，$28$的真因数有$1$、$2$、$4$、$7$、$14$。这些真因数的和恰好为$28$，因此我们称$28$是完全数。

有趣的是，$220$的真因数之和是$284$，同时$284$的真因数之和是$220$，构成了一个长度为$2$的链，我们称之为亲和数对。

不怎么为人所知的是，存在满足类似性质但更长的序列。例如，从$12496$出发，可以构成一个长度为$5$的链：

$$ 12496 \rightarrow 14288 \rightarrow 15472 \rightarrow 14536 \rightarrow 14264 (\rightarrow 12496 \rightarrow \ldots)$$

这条链最后又回到了起点，我们称之为亲和数链。

找出最长的、所有元素都不超过一百万的亲和数链，并给出其中最小的数。