The next problem up in Project Euler concerns a rather exotic-sounding kind of number pairing: the amicable pair. The problem itself contains a thorough definition and explanation of what defines such a pair:

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.

To find all amicable numbers under 10,000, I used the algorithm from my solution to problem 12 to find the divisors for all numbers between 1 and 9,999, and then searched an array constructed therefrom for amicable pairs.

```using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace Problem021
{
class Program
{
static void Main(string[] args)
{
int[] pairs = new int;
int divisors = 1;
int sumOfAmicableNumbers = 0;

for (int i = 1; i < 10000; i++)
{
for (int j = 2; j <= (int)Math.Sqrt(i); j++)
{
if (j == Math.Sqrt(i))
break;
if (i % j == 0)
divisors += j + (i / j);
}
pairs[i] = divisors;
divisors = 1;
}

for (int i = 1; i < 10000; i++)
{
if (pairs[i] < 10000)
{
if (i != pairs[i] && i == pairs[pairs[i]])
{
sumOfAmicableNumbers += i;
}
}
}

Console.WriteLine(sumOfAmicableNumbers);
}
}
}
```