# Amicable Numbers

Two positive integer numbers (m, n) are amicable if the sum of the proper divisors of (m) is equal to (n) and the sum of the proper divisors of (n) is equal to (m). A proper divisor of a number does not include the number itself. For example, the proper divisors of (4) are 1 and 2 only.

**Amicable Numbers Examples**

( 220 , 284 ) ( 1184 , 1210 ) ( 2620 , 2924 ) ( 5020 , 5564 ) ( 6232 , 6368 ) ( 10744 , 10856 ) ( 12285 , 14595 ) ( 17296 , 18416 ) ( 63020 , 76084 ) ( 66928 , 66992 )

**Amicable Numbers C++**

//Includes #include <iostream> //STD using namespace std; //Check if n and m are amicable int isAmicable(int n, int m) { //Sum of (n) proper divisors int ns = 0; //Sum of (m) proper divisors int ms = 0; //Boolean: 1 if the sum of (n) proper divisors equals to (m) int nb = 0; //Boolean: 1 if the sum of (m) proper divisors equals to (n) int mb = 0; //Find all proper divisors of (n) and calculate the sum for (int i = 1; i < n; i++) { if (n % i == 0) { ns = ns + i; } } //This must be true if the pair are amicable if (ns == m) { nb = 1; } //Find all proper divisors of (m) and calculate the sum for (int j = 1; j < m; j++) { if (m % j == 0) { ms = ms + j; } } //This must be true if the pair are amicable if (ms == n) { mb = 1; } //If both conditions are true then (n) and (m) are amicable if (nb == 1 && mb == 1) { return 1; } else { return 0; } } //Main function int main() { //Sample numbers int n = 5020; int m = 5564; //Check if they are amicable if (isAmicable(n, m) == 1) { cout << "YES : (" << n << "," << m << ")"; } else { cout << "NO : (" << n << "," << m << ")"; } return 0; }

**Amicable Numbers Under 10000**

If you want to print amicable numbers list under 10000 then you need to have two nested loops. The first loop goes from i = 1 to 10000 and the second loop goes from j = i + 1 to 1000. There is no need to go from 1 to 10000 in each loop otherwise there is going to be a lot of repetitions. Note also that these two nested loops are going to take a long time to run for larger numbers of (n) and (m).

**Amicable Numbers Program in Java**

//Main class public class Amicable { //Main function public static void main(String[] args) { //Sample numbers int n = 5020; int m = 5564; //Check if they are amicable if (isAmicable(n, m) == 1) { System.out.println("YES : (" + n + "," + m + ")"); } else { System.out.println("NO : (" + n + "," + m + ")"); } } //Check if n and m are amicable static int isAmicable(int n, int m) { //Sum of (n) proper divisors int ns = 0; //Sum of (m) proper divisors int ms = 0; //Boolean: 1 if the some of (n) proper divisors equals to (m) int nb = 0; //Boolean: 1 if the some of (m) proper divisors equals to (n) int mb = 0; //Find all proper divisors of (n) and calculate the sum for (int i = 1; i < n; i++) { if (n % i == 0) { ns = ns + i; } } //This must be true if the pair are amicable if (ns == m) { nb = 1; } //Find all proper divisors of (m) and calculate the sum for (int j = 1; j < m; j++) { if (m % j == 0) { ms = ms + j; } } //This must be true if the pair are amicable if (ms == n) { mb = 1; } //If both conditions are true then (n) and (m) are amicable if (nb == 1 && mb == 1) { return 1; } else { return 0; } } }