**Amicable numbers** or **friendly numbers** are a pair of numbers whose sum of proper divisors equals the other.

Let's go through an example to understand this better. Together, $220$ and $284$ make up the smallest pair of amicable numbers. The proper divisors of 220 are $1$, $2$, $4$, $5$, $10$, $11$, $20$, $22$, $44$, $55$, and $110$. These proper divisors add up to $284$. Similarly, the proper divisors of $284$ are $1$, $2$, $4$, $71$, and $142$, which sum up to $220$.

# Implementation

Below is a program that checks whether or not two numbers form an amicable pair. Feel free to change num1 and num2 to check if a pair is amicable.

# A function that takes an integer and returns the sum of its proper divisors. 
# E.g. proper divisors of 8 are 1, 2 and 4 so the function will return 
# 7 (1 + 2 + 4) when 8 is given as input 
def sum_of_proper_divisors(num):
    divisor_sum = 0
    for i in range(1, num):
        if (num%i == 0):
            divisor_sum += i
    return divisor_sum


# A function that takes two integers as input and returns true if the two
# form an amicable pair and false if not
def form_amicable_pair(num1, num2):
    if (sum_of_proper_divisors(num1) == num2) and (sum_of_proper_divisors(num2) == num1):
        return True
    else:
        return False

# Two integers being tested
num1 = 220
num2 = 284

if form_amicable_pair(num1, num2):
    print(str(num1)+" and "+str(num2)+" are amicable numbers!")
else:
    print(str(num1)+" and "+str(num2)+" are not amicable numbers!")


Python

#include <iostream>
using namespace std;

/* A function that takes an integer and returns the sum of its proper divisors. 
 E.g. proper divisors of 8 are 1, 2 and 4 so the function will return 
 7 (1 + 2 + 4) when 8 is given as input */
int sum_of_proper_divisors(int num)
{
  int divisor_sum = 0;
  for (int i = 1; i < num; i++)
  {
    if (num%i == 0)
      divisor_sum += i;
  }
  return divisor_sum;
}

/* A function that takes two integers as input and returns true if the two
 form an amicable pair and false if not */
bool form_amicable_pair(int num1, int num2)
{
  if ((sum_of_proper_divisors(num1) == num2) && (sum_of_proper_divisors(num2) == num1))
    return true;
  else
    return false;
}

int main() {
  // Two integers being tested
  int num1 = 220;
  int num2 = 284;

  if (form_amicable_pair(num1, num2))
    cout << num1 << " and " << num2 << " are amicable numbers!" << endl;
  else
    cout << num1 << " and " << num2 << " are not amicable numbers!" << endl;
  return 0;
}

class AmicableNumbers {
    /* A function that takes an integer and returns the sum of its proper 
     divisors. E.g. proper divisors of 8 are 1, 2 and 4 so the function will 
     return 7 (1 + 2 + 4) when 8 is given as input */
    static int sum_of_proper_divisors(int num)
    {
      int divisor_sum = 0;
      for (int i = 1; i < num; i++)
      {
        if (num%i == 0)
          divisor_sum += i;
      }
      return divisor_sum;
    }

    /* A function that takes two integers as input and returns true if the two
     form an amicable pair and false if not */
    static boolean form_amicable_pair(int num1, int num2)
    {
      if ((sum_of_proper_divisors(num1) == num2) && (sum_of_proper_divisors(num2) == num1))
        return true;
      else
        return false;
    }

    public static void main( String args[] ) {
      // Two integers being tested
      int num1 = 220;
      int num2 = 284;

      if (form_amicable_pair(num1, num2))
        System.out.println(num1 + " and " + num2 + " are amicable numbers!"); 
      else
        System.out.println(num1 + " and " + num2 + " are not amicable numbers!");
    }
}

Java

What are amicable numbers?

Numbers are amicable if each number's divisors sum to the other. Example: 220 and 284.