Pairs of Amicable Numbers

Last Updated : 21 Jan, 2025

Amicable numbers are pairs of two integers (a, b) such that:

Sum of the proper divisors of a = b and Sum of the proper divisors of b = a,

where a ≠ b.

Examples: (220,284)

Proper divisors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110
- Sum = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
Proper divisors of 284: 1, 2, 4, 71, 142
- Sum = 1 + 2 + 4 + 71 + 142 = 220

Thus, (220, 284) is a pair of amicable numbers.

Some other examples include:

(1184, 1210)
(2620, 2924)
(5020, 5564)
(6232, 6368)

Example:

Input  : arr[] = {220, 284, 1184, 1210, 2, 5}
Output : 2
Explanation : (220, 284) and (1184, 1210) form amicable pair.

Input  : arr[] = {2620, 2924, 5020, 5564, 6232, 6368}
Output : 3
Explanation : (2620, 2924), (5020, 5564) and (6232, 6368) forms amicable pair.

A simple solution is to traverse each pair and check if they form an amicable pair, if they do we increment the count.

Implementation:

C++

// A simple C++ program to count 
// amicable pairs in an array.
#include <bits/stdc++.h>
using namespace std;

// Calculate the sum
// of proper divisors
int sumOfDiv(int x)
{
    // 1 is a proper divisor
    int sum = 1;
    for (int i = 2; i <= sqrt(x); i++) 
    {
        if (x % i == 0) 
        {
            sum += i;

            // To handle perfect squares
            if (x / i != i)
                sum += x / i;
        }
    }
    return sum;
}

// Check if pair is amicable
bool isAmicable(int a, int b)
{
    return (sumOfDiv(a) == b && 
            sumOfDiv(b) == a);
}

// This function prints pair 
// of amicable pairs present 
// in the input array
int countPairs(int arr[], int n)
{
    int count = 0;

    // Iterate through each 
    // pair, and find if it
    // an amicable pair
    for (int i = 0; i < n; i++)
        for (int j = i + 1; j < n; j++)
            if (isAmicable(arr[i], arr[j]))
                count++;

    return count;
}

// Driver code
int main()
{
    int arr1[] = { 220, 284, 1184, 
                   1210, 2, 5 };
    int n1 = sizeof(arr1) / sizeof(arr1[0]);
    cout << countPairs(arr1, n1) 
         << endl;

    int arr2[] = { 2620, 2924, 5020, 
                   5564, 6232, 6368 };
    int n2 = sizeof(arr2) / sizeof(arr2[0]);
    cout << countPairs(arr2, n2);
    return 0;
}

Java

// A simple Java program to count
// amicable pairs in an array.
import java.io.*;

class GFG 
{

    // Calculate the sum 
    // of proper divisors
    static int sumOfDiv(int x)
    {

        // 1 is a proper divisor
        int sum = 1;
        for (int i = 2; i <= Math.sqrt(x); i++) 
        {
            if (x % i == 0) 
            {
                sum += i;

                // To handle perfect squares
                if (x / i != i)
                    sum += x / i;
            }
        }

        return sum;
    }

    // Check if pair is amicable
    static boolean isAmicable(int a, int b)
    {
        return (sumOfDiv(a) == b && 
                sumOfDiv(b) == a);
    }

    // This function prints pair
    //  of amicable pairs present
    // in the input array
    static int countPairs(int arr[], int n)
    {
        int count = 0;

        // Iterate through each pair, 
        // and find if it an amicable pair
        for (int i = 0; i < n; i++)
            for (int j = i + 1; j < n; j++)
                if (isAmicable(arr[i], arr[j]))
                    count++;

        return count;
    }

    // Driver code
    public static void main(String args[])
    {

        int arr1[] = { 220, 284, 1184, 
                       1210, 2, 5 };
        int n1 = arr1.length;

        System.out.println(countPairs(arr1, n1));

        int arr2[] = { 2620, 2924, 5020,
                       5564, 6232, 6368 };
        int n2 = arr2.length;

        System.out.println(countPairs(arr2, n2));
    }
}

// This code is contributed by Anshika Goyal.

Python

# Python3 program to count 
# amicable pairs in an array

# Calculate the sum 
# of proper divisors
def sumOfDiv(x):
    sum = 1
    for i in range(2, x):
        if x % i == 0:
            sum += i
    return sum

# Check if pair is amicable
def isAmicable(a, b):
    if sumOfDiv(a) == b and sumOfDiv(b) == a:
        return True
    else:
        return False

# This function prints pair 
# of amicable pairs present 
# in the input array
def countPairs(arr, n):
    count = 0
    for i in range(0, n):
        for j in range(i + 1, n):
            if isAmicable(arr[i], arr[j]):
                count = count + 1
    return count

# Driver Code
arr1 = [220, 284, 1184,
        1210, 2, 5]
n1 = len(arr1)
print(countPairs(arr1, n1))

arr2 = [2620, 2924, 5020, 
        5564, 6232, 6368]
n2 = len(arr2)
print(countPairs(arr2, n2))

# This code is contributed 
# by Smitha Dinesh Semwal

// A simple C# program to count 
// amicable pairs in an array.
using System;

class GFG 
{
    
    // Calculate the sum
    // of proper divisors
    static int sumOfDiv(int x)
    {
        
        // 1 is a proper divisor
        int sum = 1;
        for (int i = 2; i <= Math.Sqrt(x); i++) 
        {
            if (x % i == 0) 
            {
                sum += i;

                // To handle perfect squares
                if (x / i != i)
                    sum += x / i;
            }
        }
        
        return sum;
    }

    // Check if pair is amicable
    static bool isAmicable(int a, int b)
    {
        return (sumOfDiv(a) == b &&
                sumOfDiv(b) == a);
    }

    // This function prints pair
    // of amicable pairs present 
    // in the input array
    static int countPairs(int []arr, int n)
    {
        int count = 0;

        // Iterate through each pair, 
        // and find if it an amicable pair
        for (int i = 0; i < n; i++)
        
            for (int j = i + 1; j < n; j++)
                if (isAmicable(arr[i], arr[j]))
                    count++;

        return count;
    }

    // Driver code
    public static void Main() 
    {

        int []arr1 = {220, 284, 1184, 
                      1210, 2, 5};
        int n1 = arr1.Length;
        
        Console.WriteLine(countPairs(arr1, n1));

        int []arr2 = {2620, 2924, 5020, 
                      5564, 6232, 6368};
        int n2 = arr2.Length;

        Console.WriteLine(countPairs(arr2, n2));
    }
}

// This code is contributed by vt_m.

JavaScript

<script>
    // A simple Javascript program to count 
    // amicable pairs in an array.
    
    // Calculate the sum
    // of proper divisors
    function sumOfDiv(x)
    {
          
        // 1 is a proper divisor
        let sum = 1;
        for (let i = 2; i <= Math.sqrt(x); i++) 
        {
            if (x % i == 0) 
            {
                sum += i;
  
                // To handle perfect squares
                if (parseInt(x / i, 10) != i)
                    sum += parseInt(x / i, 10);
            }
        }
          
        return sum;
    }
  
    // Check if pair is amicable
    function isAmicable(a, b)
    {
        return (sumOfDiv(a) == b &&
                sumOfDiv(b) == a);
    }
  
    // This function prints pair
    // of amicable pairs present 
    // in the input array
    function countPairs(arr, n)
    {
        let count = 0;
  
        // Iterate through each pair, 
        // and find if it an amicable pair
        for (let i = 0; i < n; i++)
          
            for (let j = i + 1; j < n; j++)
                if (isAmicable(arr[i], arr[j]))
                    count++;
  
        return count;
    }
    
    let arr1 = [220, 284, 1184, 1210, 2, 5];
    let n1 = arr1.length;

    document.write(countPairs(arr1, n1) + "</br>");

    let arr2 = [2620, 2924, 5020, 5564, 6232, 6368];
    let n2 = arr2.length;

    document.write(countPairs(arr2, n2));
    
</script>

PHP

<?php
// A simple PHP program to count 
// amicable pairs in an array.

// Calculate the sum 
// of proper divisors
function sumOfDiv( $x)
{
    // 1 is a proper divisor
    $sum = 1;
    for ( $i = 2; $i <= sqrt($x); $i++) 
    {
        if ($x % $i == 0) 
        {
            $sum += $i;

            // To handle perfect squares
            if ($x / $i != $i)
                $sum += $x / $i;
        }
    }
    return $sum;
}

// Check if pair is amicable
function isAmicable( $a, $b)
{
    return (sumOfDiv($a) == $b and 
            sumOfDiv($b) == $a);
}

// This function prints pair 
// of amicable pairs present 
// in the input array
function countPairs( $arr, $n)
{
    $count = 0;

    // Iterate through each pair, 
    // and find if it an amicable pair
    for ( $i = 0; $i < $n; $i++)
        for ( $j = $i + 1; $j < $n; $j++)
            if (isAmicable($arr[$i], $arr[$j]))
                $count++;

    return $count;
}

// Driver code
$arr1 = array( 220, 284, 1184, 
               1210, 2, 5 );
$n1 = count($arr1);
echo countPairs($arr1, $n1),"\n";

$arr2 = array( 2620, 2924, 5020, 
               5564, 6232, 6368 );
$n2 = count($arr2);
echo countPairs($arr2, $n2);

// This code is contributed by anuj_67.
?>

Output

2
3

An efficient solution is to keep the numbers stored in a map and for every number, we find the sum of its proper divisor and check if that's also present in the array. If it is present, we can check if they form an amicable pair or not.

Thus, the complexity would be considerably reduced. Below is the C++ program for the same.

Implementation:

C++

// Efficient C++ program to count 
// Amicable pairs in an array.
#include <bits/stdc++.h>
using namespace std;

// Calculate the sum 
// of proper divisors
int sumOfDiv(int x)
{
    // 1 is a proper divisor
    int sum = 1;
    for (int i = 2; i <= sqrt(x); i++) 
    {
        if (x % i == 0) {
            sum += i;

            // To handle perfect squares
            if (x / i != i)
                sum += x / i;
        }
    }
    return sum;
}

// Check if pair is amicable
bool isAmicable(int a, int b)
{
    return (sumOfDiv(a) == b && 
            sumOfDiv(b) == a);
}

// This function prints count 
// of amicable pairs present 
// in the input array
int countPairs(int arr[], int n)
{
    // Map to store the numbers
    unordered_set<int> s;
    int count = 0;
    for (int i = 0; i < n; i++)
        s.insert(arr[i]);

    // Iterate through each number, 
    // and find the sum of proper 
    // divisors and check if it's 
    // also present in the array
    for (int i = 0; i < n; i++) 
    {
        if (s.find(sumOfDiv(arr[i])) != s.end()) 
        {
            // It's sum of proper divisors
            int sum = sumOfDiv(arr[i]);
            if (isAmicable(arr[i], sum))
                count++;
        }
    }

    // As the pairs are counted 
    // twice, thus divide by 2
    return count / 2;
}

// Driver code
int main()
{
    int arr1[] = { 220, 284, 1184, 
                   1210, 2, 5 };
    int n1 = sizeof(arr1) / sizeof(arr1[0]);
    cout << countPairs(arr1, n1) 
         << endl;
    
    int arr2[] = { 2620, 2924, 5020, 
                   5564, 6232, 6368 };
    int n2 = sizeof(arr2) / sizeof(arr2[0]);
    cout << countPairs(arr2, n2) 
         << endl;
    return 0;
}

Java

// Efficient Java program to count 
// Amicable pairs in an array.
import java.util.*;

class GFG 
{

// Calculate the sum 
// of proper divisors
static int sumOfDiv(int x)
{
    // 1 is a proper divisor
    int sum = 1;
    for (int i = 2; i <= Math.sqrt(x); i++) 
    {
        if (x % i == 0) 
        {
            sum += i;

            // To handle perfect squares
            if (x / i != i)
                sum += x / i;
        }
    }
    return sum;
}

// Check if pair is amicable
static boolean isAmicable(int a, int b)
{
    return (sumOfDiv(a) == b && 
            sumOfDiv(b) == a);
}

// This function prints count 
// of amicable pairs present 
// in the input array
static int countPairs(int arr[], int n)
{
    // Map to store the numbers
    HashSet<Integer> s = new HashSet<Integer>();
    int count = 0;
    for (int i = 0; i < n; i++)
        s.add(arr[i]);

    // Iterate through each number, 
    // and find the sum of proper 
    // divisors and check if it's 
    // also present in the array
    for (int i = 0; i < n; i++) 
    {
        if (s.contains(sumOfDiv(arr[i]))) 
        {
            // It's sum of proper divisors
            int sum = sumOfDiv(arr[i]);
            if (isAmicable(arr[i], sum))
                count++;
        }
    }

    // As the pairs are counted 
    // twice, thus divide by 2
    return count / 2;
}

// Driver code
public static void main(String[] args) 
{
    int arr1[] = { 220, 284, 1184, 
                   1210, 2, 5 };
    int n1 = arr1.length;
    System.out.println(countPairs(arr1, n1));
    
    int arr2[] = { 2620, 2924, 5020, 
                   5564, 6232, 6368 };
    int n2 = arr2.length;
    System.out.println(countPairs(arr2, n2));
}
}

// This code is contributed by PrinciRaj1992

Python

# Efficient Python3 program to count 
# Amicable pairs in an array.
import math

# Calculating the sum
# of proper divisors
def sumOfDiv(x): 
    
    # 1 is a proper divisor
    sum = 1;
    for i in range(2,int(math.sqrt(x))):
        if x % i==0:
            sum += i
            
            # To handle perfect squares
            if i != x/i:
                sum += x/i
    return int(sum);

# check if pair is amicable
def isAmicable(a, b):
    return (sumOfDiv(a) == b and
            sumOfDiv(b) == a)

# This function prints count 
# of amicable pairs present 
# in the input array
def countPairs(arr,n): 
    
    # Map to store the numbers
    s = set()
    count = 0
    for i in range(n):
        s.add(arr[i])
    
    # Iterate through each number, 
    # and find the sum of proper 
    # divisors and check if it's 
    # also present in the array
    for i in range(n):     
        if sumOfDiv(arr[i]) in s:
            
            # It's sum of proper divisors
            sum = sumOfDiv(arr[i])
            if isAmicable(arr[i], sum):
                count += 1        
    
    # As the pairs are counted 
    # twice, thus divide by 2
    return int(count/2);
    
# Driver Code 
arr1 = [220, 284, 1184, 
        1210, 2, 5]
n1 = len(arr1)
print(countPairs(arr1, n1)) 

arr2 = [2620, 2924, 5020, 
        5564, 6232, 6368]
n2 = len(arr2)
print(countPairs(arr2, n2)) 

# This code is contributed 
# by Naveen Babbar

// Efficient C# program to count 
// Amicable pairs in an array.
using System;
using System.Collections.Generic;
    
class GFG 
{

// Calculate the sum 
// of proper divisors
static int sumOfDiv(int x)
{
    // 1 is a proper divisor
    int sum = 1;
    for (int i = 2; i <= Math.Sqrt(x); i++) 
    {
        if (x % i == 0) 
        {
            sum += i;

            // To handle perfect squares
            if (x / i != i)
                sum += x / i;
        }
    }
    return sum;
}

// Check if pair is amicable
static Boolean isAmicable(int a, int b)
{
    return (sumOfDiv(a) == b && 
            sumOfDiv(b) == a);
}

// This function prints count 
// of amicable pairs present 
// in the input array
static int countPairs(int []arr, int n)
{
    // Map to store the numbers
    HashSet<int> s = new HashSet<int>();
    int count = 0;
    for (int i = 0; i < n; i++)
        s.Add(arr[i]);

    // Iterate through each number, 
    // and find the sum of proper 
    // divisors and check if it's 
    // also present in the array
    for (int i = 0; i < n; i++) 
    {
        if (s.Contains(sumOfDiv(arr[i]))) 
        {
            // It's sum of proper divisors
            int sum = sumOfDiv(arr[i]);
            if (isAmicable(arr[i], sum))
                count++;
        }
    }

    // As the pairs are counted 
    // twice, thus divide by 2
    return count / 2;
}

// Driver code
public static void Main(String[] args) 
{
    int []arr1 = { 220, 284, 1184, 
                   1210, 2, 5 };
    int n1 = arr1.Length;
    Console.WriteLine(countPairs(arr1, n1));
    
    int []arr2 = { 2620, 2924, 5020, 
                   5564, 6232, 6368 };
    int n2 = arr2.Length;
    Console.WriteLine(countPairs(arr2, n2));
}
}

// This code is contributed by Princi Singh

JavaScript

<script>

//  JavaScript program to count
// Amicable pairs in an array.

// Calculate the sum
// of proper divisors
function sumOfDiv(x)
{
    // 1 is a proper divisor
    let sum = 1;
    for (let i = 2; i <= Math.sqrt(x); i++)
    {
        if (x % i == 0)
        {
            sum += i;
 
            // To handle perfect squares
            if (x / i != i)
                sum += x / i;
        }
    }
    return sum;
}
 
// Check if pair is amicable
function isAmicable(a, b)
{
    return (sumOfDiv(a) == b &&
            sumOfDiv(b) == a);
}
 
// This function prints count
// of amicable pairs present
// in the input array
function countPairs(arr, n)
{
    // Map to store the numbers
    let s = new Set();
    let count = 0;
    for (let i = 0; i < n; i++)
        s.add(arr[i]);
 
    // Iterate through each number,
    // and find the sum of proper
    // divisors and check if it's
    // also present in the array
    for (let i = 0; i < n; i++)
    {
        if (s.has(sumOfDiv(arr[i])))
        {
            // It's sum of proper divisors
            let sum = sumOfDiv(arr[i]);
            if (isAmicable(arr[i], sum))
                count++;
        }
    }
 
    // As the pairs are counted
    // twice, thus divide by 2
    return  Math.floor(count / 2);
}
    
    // Driver code     
    
    let arr1 = [ 220, 284, 1184,
                   1210, 2, 5 ];
    let n1 = arr1.length;
    document.write(countPairs(arr1, n1) + "<br/>");
     
    let arr2 = [ 2620, 2924, 5020,
                   5564, 6232, 6368 ];
    let n2 = arr2.length;
    document.write(countPairs(arr2, n2) + "<br/>");
     
</script>