# Project Euler

## Amicable numbers

### Problem 21

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 ab, 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.

This problem can modify the function built for problem 12 to find a divisor of a number to return the sum of the divisors of a number. Notice this divisor sum function should only check up to the $\sqrt{n}$ because any divisor found below the $\sqrt{n}$ will have a corresponding divisor above $\sqrt{n}$ and simply obtained by dividing the first divisor found into the number the divisor came from. Of course, a perfect square need to count the value $\sqrt{n}$ only once. Once a divisor sum function is presented, to check $d(i) = d(d(i)) | d(i) \neq i$, an array is setup to remember which number has a corresponding amicable numbers to add these number up at the end and eliminate the need to recalculate any i.