Problem 21

Amicable numbers

Let $d(n)$ be defined as the sum of proper divisors of $n$ (numbers less than $n$ which divide evenly into $n$).

If $d(a) = b$ and $d(b) = a$, where $a \neq b$, then $a$ and $b$ are an amicable pair and each of $a$ and $b$ are called amicable numbers.

For example, the proper divisors of $220$ are $1$, $2$, $4$, $5$, $10$, $11$, $20$, $22$, $44$, $55$ and $110$; therefore $d(220) = 284$. The proper divisors of $284$ are $1$, $2$, $4$, $71$ and $142$; so $d(284) = 220$.

Evaluate the sum of all the amicable numbers under $10000$.

亲和数

记$d(n)$为$n$的所有真约数（小于$n$且整除$n$的正整数）之和。

如果$d(a) = b$，$d(b) = a$，而且$a\neq b$，那么$a$和$b$构成一个亲和数对，$a$和$b$都被称为亲和数。

例如，$220$的真因数包括$1$、$2$、$4$、$5$、$10$、$11$、$20$、$22$、$44$、$55$和$110$，因此$d(220) = 284$；而$284$的真因数包括$1$、$2$、$4$、$71$和$142$，因此$d(284) = 220$。

求所有小于$10000$的亲和数之和。