Sum of divisors of factorial of a number

Given a number n, we need to calculate the sum of divisors of factorial of the number.

Examples:

Input : 4
Output : 60
Factorial of 4 is 24. Divisors of 24 are
1 2 3 4 6 8 12 24, sum of these is 60.

Input : 6
Output : 2418

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

A Simple Solution is to first compute factorial of given number, then count number divisors of the factorial. This solution is not efficient and may cause overflow due to factorial computation.

Below is the implementation of above approach:

C++

// C++ program to find sum of proper divisor of  
// factorial of a number  
#include <bits/stdc++.h> 
using namespace std; 
  
// function to calculate factorial  
int fact(int n) 
{ 
    if (n == 0) 
        return 1; 
    return n * fact(n - 1); 
} 
  
// function to calculate sum of divisor  
int div(int x) 
{ 
    int ans = 0; 
    for (int i = 1; i<= x; i++) 
        if (x % i == 0) 
            ans += i; 
    return ans; 
} 
  
// Returns sum of divisors of n!  
int sumFactDiv(int n) 
{ 
    return div(fact(n)); 
} 
  
// Driver Code 
int main() 
{ 
    int n = 4; 
    cout << sumFactDiv(n); 
} 
  
// This code is contributed  
// by Akanksha Rai 

C

// C program to find sum of proper divisor of  
// factorial of a number  
#include <stdio.h> 
// function to calculate factorial  
  
int fact(int n) { 
    if (n == 0) 
        return 1; 
    return n * fact(n - 1); 
} 
  
// function to calculate sum of divisor  
  
int div(int x) { 
    int ans = 0; 
    for (int i = 1; i<= x; i++) 
        if (x % i == 0) 
            ans += i; 
    return ans; 
} 
  
// Returns sum of divisors of n!  
  
int sumFactDiv(int n) { 
    return div(fact(n)); 
} 
  
// driver program  
  
int main() { 
    int n = 4; 
    printf("%d",sumFactDiv(n)); 
} 

Java

// Java program to find sum of proper divisor of 
// factorial of a number 
import java.io.*; 
import java.util.*; 
  
public class Division 
{ 
    // function to calculate factorial 
    static int fact(int n) 
    { 
        if (n == 0) 
            return 1; 
        return n*fact(n-1); 
    } 
  
    // function to calculate sum of divisor 
    static int div(int x) 
    { 
        int ans = 0; 
        for (int i = 1; i<= x; i++) 
            if (x%i == 0) 
                ans += i; 
        return ans; 
    } 
  
    // Returns sum of divisors of n! 
    static int sumFactDiv(int n) 
    { 
        return div(fact(n)); 
    } 
  
    // driver program 
    public static void main(String args[]) 
    { 
        int n = 4; 
        System.out.println(sumFactDiv(n)); 
    } 
}  

Python3

# Python 3 program to find sum of proper  
# divisor of factorial of a number  
  
# function to calculate factorial  
def fact(n): 
      
    if (n == 0): 
        return 1
    return n * fact(n - 1) 
  
# function to calculate sum  
# of divisor  
def div(x): 
    ans = 0; 
    for i in range(1, x + 1): 
        if (x % i == 0): 
            ans += i 
    return ans 
  
# Returns sum of divisors of n!  
def sumFactDiv(n): 
    return div(fact(n)) 
  
# Driver Code 
n = 4
print(sumFactDiv(n)) 
  
# This code is contributed 
# by Rajput-Ji 

C#

// C# program to find sum of proper  
// divisor of factorial of a number 
using System; 
class Division { 
      
    // function to calculate factorial 
    static int fac(int n) 
    { 
        if (n == 0) 
            return 1; 
        return n * fac(n - 1); 
    } 
  
    // function to calculate 
    // sum of divisor 
    static int div(int x) 
    { 
        int ans = 0; 
        for (int i = 1; i <= x; i++) 
            if (x % i == 0) 
                ans += i; 
        return ans; 
    } 
  
    // Returns sum of divisors of n! 
    static int sumFactDiv(int n) 
    { 
        return div(fac(n)); 
    } 
  
    // Driver Code 
    public static void Main() 
    { 
        int n = 4; 
        Console.Write(sumFactDiv(n)); 
    } 
}  
  
// This code is contributed by Nitin Mittal. 

PHP

<?php 
// PHP program to find sum of proper  
// divisor of factorial of a number  
  
// function to calculate factorial  
function fact($n) 
{ 
    if ($n == 0) 
        return 1; 
    return $n * fact($n - 1); 
} 
  
// function to calculate sum of divisor  
function div($x) 
{ 
    $ans = 0; 
    for ($i = 1; $i<= $x; $i++) 
        if ($x % $i == 0) 
            $ans += $i; 
    return $ans; 
} 
  
// Returns sum of divisors of n!  
function sumFactDiv($n) 
{ 
    return div(fact($n)); 
} 
  
// Driver Code 
$n = 4; 
echo sumFactDiv($n); 
  
// This code is contributed  
// by Akanksha Rai 
?> 

Output :

An efficient solution is based on Legendre’s formula. Below are the steps.

Find all prime numbers less than or equal to n (input number). We can use Sieve Algorithm for this. Let n be 6. All prime numbers less than 6 are {2, 3, 5}.
For each prime number p find the largest power of it that divides n!. We use below Legendre’s formula formula for this purpose.

The value of largest power that divides p is floor value of each term n/p + n/(p2) + n/(p3) + ……
Let these values be exp1, exp2, exp3, .. Using the above formula, we get below values for n = 6.

The largest power of 2 that divides 6!, exp1 = 4.
The largest power of 3 that divides 6!, exp2 = 2.
The largest power of 5 that divides 6!, exp3 = 1.

The result is based on the Divisor Function.

C++

// C++ program to find sum of divisors in n! 
#include<bits/stdc++.h> 
#include<math.h> 
using namespace std; 
  
// allPrimes[] stores all prime numbers less 
// than or equal to n. 
vector<int> allPrimes; 
  
// Fills above vector allPrimes[] for a given n 
void sieve(int n) 
{ 
    // Create a boolean array "prime[0..n]" and 
    // initialize all entries it as true. A value 
    // in prime[i] will finally be false if i is 
    // not a prime, else true. 
    vector<bool> prime(n+1, true); 
  
    // Loop to update prime[] 
    for (int p = 2; p*p <= n; p++) 
    { 
        // If prime[p] is not changed, then it 
        // is a prime 
        if (prime[p] == true) 
        { 
            // Update all multiples of p 
            for (int i = p*2; i <= n; i += p) 
                prime[i] = false; 
        } 
    } 
  
    // Store primes in the vector allPrimes 
    for (int p = 2; p <= n; p++) 
        if (prime[p]) 
            allPrimes.push_back(p); 
} 
  
// Function to find all result of factorial number 
int factorialDivisors(int n) 
{ 
    sieve(n);  // create sieve 
  
    // Initialize result 
    int result = 1; 
  
    // find exponents of all primes which divides n 
    // and less than n 
    for (int i = 0; i < allPrimes.size(); i++) 
    { 
        // Current divisor 
        int p = allPrimes[i]; 
  
        // Find the highest power (stored in exp)' 
        // of allPrimes[i] that divides n using 
        // Legendre's formula. 
        int exp = 0; 
        while (p <= n) 
        { 
            exp = exp + (n/p); 
            p = p*allPrimes[i]; 
        } 
  
        // Using the divisor function to calculate 
        // the sum 
        result = result*(pow(allPrimes[i], exp+1)-1)/ 
                                    (allPrimes[i]-1); 
    } 
  
    // return total divisors 
    return result; 
} 
  
// Driver program to run the cases 
int main() 
{ 
    cout << factorialDivisors(4); 
    return 0; 
} 

Java

// Java program to find sum of divisors in n!  
import java.util.*; 
  
class GFG{ 
// allPrimes[] stores all prime numbers less  
// than or equal to n.  
static ArrayList<Integer> allPrimes=new ArrayList<Integer>();  
  
// Fills above vector allPrimes[] for a given n  
static void sieve(int n)  
{  
    // Create a boolean array "prime[0..n]" and  
    // initialize all entries it as true. A value  
    // in prime[i] will finally be false if i is  
    // not a prime, else true.  
    boolean[] prime=new boolean[n+1];  
  
    // Loop to update prime[]  
    for (int p = 2; p*p <= n; p++)  
    {  
        // If prime[p] is not changed, then it  
        // is a prime  
        if (prime[p] == false)  
        {  
            // Update all multiples of p  
            for (int i = p*2; i <= n; i += p)  
                prime[i] = true;  
        }  
    }  
  
    // Store primes in the vector allPrimes  
    for (int p = 2; p <= n; p++)  
        if (prime[p]==false)  
            allPrimes.add(p);  
}  
  
// Function to find all result of factorial number  
static int factorialDivisors(int n)  
{  
    sieve(n); // create sieve  
  
    // Initialize result  
    int result = 1;  
  
    // find exponents of all primes which divides n  
    // and less than n  
    for (int i = 0; i < allPrimes.size(); i++)  
    {  
        // Current divisor  
        int p = allPrimes.get(i);  
  
        // Find the highest power (stored in exp)'  
        // of allPrimes[i] that divides n using  
        // Legendre's formula.  
        int exp = 0;  
        while (p <= n)  
        {  
            exp = exp + (n/p);  
            p = p*allPrimes.get(i);  
        }  
  
        // Using the divisor function to calculate  
        // the sum  
        result = result*((int)Math.pow(allPrimes.get(i), exp+1)-1)/  
                                    (allPrimes.get(i)-1);  
    }  
  
    // return total divisors  
    return result;  
}  
  
// Driver program to run the cases  
public static void main(String[] args)  
{  
    System.out.println(factorialDivisors(4));  
}  
} 
// This code is contributed by mits 

Python3

# Python3 program to find sum of divisors in n!

# allPrimes[] stores all prime numbers
# less than or equal to n.
allPrimes = [];

# Fills above vector allPrimes[]
# for a given n
def sieve(n):

# Create a boolean array “prime[0..n]”
# and initialize all entries it as true.
# A value in prime[i] will finally be
# false if i is not a prime, else true.
prime = [True] * (n + 1);

# Loop to update prime[]
p = 2;
while (p * p <= n): # If prime[p] is not changed, # then it is a prime if (prime[p] == True): # Update all multiples of p for i in range(p * 2, n + 1, p): prime[i] = False; p += 1; # Store primes in the vector allPrimes for p in range(2, n + 1): if (prime[p]): allPrimes.append(p); # Function to find all result of factorial number def factorialDivisors(n): sieve(n); # create sieve # Initialize result result = 1; # find exponents of all primes which # divides n and less than n for i in range(len(allPrimes)): # Current divisor p = allPrimes[i]; # Find the highest power (stored in exp)' # of allPrimes[i] that divides n using # Legendre's formula. exp = 0; while (p <= n): exp = exp + int(n / p); p = p * allPrimes[i]; # Using the divisor function to # calculate the sum result = int(result * (pow(allPrimes[i], exp + 1) - 1) / (allPrimes[i] - 1)); # return total divisors return result; # Driver Code print(factorialDivisors(4)); # This code is contributed by mits [tabby title="C#"]

// C# program to find sum of divisors in n!  
using System; 
using System.Collections; 
  
class GFG{ 
// allPrimes[] stores all prime numbers less  
// than or equal to n.  
static ArrayList allPrimes=new ArrayList();  
  
// Fills above vector allPrimes[] for a given n  
static void sieve(int n)  
{  
    // Create a boolean array "prime[0..n]" and  
    // initialize all entries it as true. A value  
    // in prime[i] will finally be false if i is  
    // not a prime, else true.  
    bool[] prime=new bool[n+1];  
  
    // Loop to update prime[]  
    for (int p = 2; p*p <= n; p++)  
    {  
        // If prime[p] is not changed, then it  
        // is a prime  
        if (prime[p] == false)  
        {  
            // Update all multiples of p  
            for (int i = p*2; i <= n; i += p)  
                prime[i] = true;  
        }  
    }  
  
    // Store primes in the vector allPrimes  
    for (int p = 2; p <= n; p++)  
        if (prime[p]==false)  
            allPrimes.Add(p);  
}  
  
// Function to find all result of factorial number  
static int factorialDivisors(int n)  
{  
    sieve(n); // create sieve  
  
    // Initialize result  
    int result = 1;  
  
    // find exponents of all primes which divides n  
    // and less than n  
    for (int i = 0; i < allPrimes.Count; i++)  
    {  
        // Current divisor  
        int p = (int)allPrimes[i];  
  
        // Find the highest power (stored in exp)'  
        // of allPrimes[i] that divides n using  
        // Legendre's formula.  
        int exp = 0;  
        while (p <= n)  
        {  
            exp = exp + (n/p);  
            p = p*(int)allPrimes[i];  
        }  
  
        // Using the divisor function to calculate  
        // the sum  
        result = result*((int)Math.Pow((int)allPrimes[i], exp+1)-1)/  
                                    ((int)allPrimes[i]-1);  
    }  
  
    // return total divisors  
    return result;  
}  
  
// Driver program to run the cases  
static void Main()  
{  
    Console.WriteLine(factorialDivisors(4));  
}  
} 
// This code is contributed by mits 

PHP

<?php 
// PHP program to find sum of divisors in n!  
  
// allPrimes[] stores all prime numbers less  
// than or equal to n.  
$allPrimes=array();  
  
// Fills above vector allPrimes[] for a given n  
function sieve($n)  
{ 
    global $allPrimes; 
    // Create a boolean array "prime[0..n]" and  
    // initialize all entries it as true. A value  
    // in prime[i] will finally be false if i is  
    // not a prime, else true.  
    $prime=array_fill(0,$n+1,true);  
  
    // Loop to update prime[]  
    for ($p = 2; $p*$p <= $n; $p++)  
    {  
        // If prime[p] is not changed, then it  
        // is a prime  
        if ($prime[$p] == true)  
        {  
            // Update all multiples of p  
            for ($i = $p*2; $i <= $n; $i += $p)  
                $prime[$i] = false;  
        }  
    }  
  
    // Store primes in the vector allPrimes  
    for ($p = 2; $p <= $n; $p++)  
        if ($prime[$p])  
            array_push($allPrimes,$p);  
}  
  
// Function to find all result of factorial number  
function factorialDivisors($n)  
{ 
    global $allPrimes; 
    sieve($n); // create sieve  
  
    // Initialize result  
    $result = 1;  
  
    // find exponents of all primes which divides n  
    // and less than n  
    for ($i = 0; $i < count($allPrimes); $i++)  
    {  
        // Current divisor  
        $p = $allPrimes[$i];  
  
        // Find the highest power (stored in exp)'  
        // of allPrimes[i] that divides n using  
        // Legendre's formula.  
        $exp = 0;  
        while ($p <= $n)  
        {  
            $exp = $exp + (int)($n/$p);  
            $p = $p*$allPrimes[$i];  
        }  
  
        // Using the divisor function to calculate  
        // the sum  
        $result = $result*(pow($allPrimes[$i], $exp+1)-1)/  
                                    ($allPrimes[$i]-1);  
    }  
  
    // return total divisors  
    return $result;  
}  
  
// Driver program to run the cases  
  
    print(factorialDivisors(4)); 
  
  
// This code is contributed by mits 
?> 

Output:

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