Two numbers are amicable numbers if they are different and the sum of each number's proper divisors equals the other number. For example, the divisors of 220 are {1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110} and 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. Also, the divisors of 284 are {1, 2, 4, 71, 142} and 1 + 2 + 4 + 71 + 142 = 220. That means 220 and 284 are amicable numbers.

Write a program that finds amicable numbers between 1 and a specified maximum.