Two numbers are said to be amicable if the sum of the proper divisors of one number is equal to that of the other number. The proper divisors of a number are the positive prime factors of the number other than the number itself. Amicable numbers should not be confused with friendly numbers. For instance, the number 220 has the proper divisors 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, whose sum is 284. The proper divisors of 284 are 1, 2, 4, 71, and 142; their sum is 220. Therefore, the numbers 220 and 284 are said to be amicable.

The solution to this problem is to iterate through all the numbers up to the given limit. For each number, compute the sum of its proper divisors. Let’s call this sum1. Repeat the process and compute the sum of the proper divisors of sum1. If the result is equal to the original number, then the number and sum1 are amicable numbers:

void print_amicables(int const limit)
{
   for (int number = 4; number < limit; ++number)
   {
      auto sum1 = sum_proper_divisors(number);
      if (sum1 < limit)
      {
         auto sum2 = sum_proper_divisors(sum1);
         if (sum2 == number && number != sum1)
         {
            std::cout << number << "," << sum1 << std::endl;
         }
      }
   }
}

In the above sample, sum_proper_divisors() is the function seen in the solution to the abundant numbers problem.