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Count Divisors of Factorial
  • Difficulty Level : Medium
  • Last Updated : 19 Jan, 2021

Given a number n, count the total number of divisors of n!.

Examples: 

Input : n = 4
Output: 8
Explanation:
4! is 24. Divisors of 24 are 1, 2, 3, 4, 6,
8, 12 and 24.

Input : n = 5
Output : 16
Explanation:
5! is 120. Divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24 30, 40, 60 and 12 

A Simple Solution is to first compute the factorial of the given number, then count the number of divisors of the factorial. This solution is not efficient and may cause overflow due to factorial computation.
A better solution is based on Legendre’s formula. Below are the step:



  1. 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}.
  2. For each prime number, p find the largest power of it that divides n!. We use Legendre’s formula for this purpose. 
    The value of largest power that divides p is ⌊n/p⌋ + ⌊n/(p2)⌋ + ⌊n/(p3)⌋ + …… 
    Let these values be exp1, exp2, exp3,… Using the above formula, we get the 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.
  3. The result is (exp1 + 1) * (exp2 + 1) * (exp3 + 1) … for all prime numbers, For n = 6, the values exp1, exp2, and exp3 are 4 2 and 1 respectively (computed in above step 2). So our result is (4 + 1)*(2 + 1) * (1 + 1) = 30

Below is the implementation of the above idea. 

C++




// C++ program to find count of divisors in n!
#include<bits/stdc++.h>
using namespace std;
typedef unsigned long long int ull;
 
// allPrimes[] stores all prime numbers less
// than or equal to n.
vector<ull> 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
ull factorialDivisors(ull n)
{
    sieve(n);  // create sieve
 
    // Initialize result
    ull result = 1;
 
    // find exponents of all primes which divides n
    // and less than n
    for (int i=0; i < allPrimes.size(); i++)
    {
        // Current divisor
        ull p = allPrimes[i];
 
        // Find the highest power (stored in exp)'
        // of allPrimes[i] that divides n using
        // Legendre's formula.
        ull exp = 0;
        while (p <= n)
        {
            exp = exp + (n/p);
            p = p*allPrimes[i];
        }
 
        // Multiply exponents of all primes less
        // than n
        result = result*(exp+1);
    }
 
    // return total divisors
    return result;
}
 
// Driver code
int main()
{
    cout << factorialDivisors(6);
    return 0;
}

Java




// JAVA program to find count of divisors in n!
 
import java.util.*;
class GFG
{
    // allPrimes[] stores all prime numbers less
    // than or equal to n.
    static Vector<Integer> allPrimes=new Vector<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];
         
        for(int i=0;i<=n;i++)
        prime[i]=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.add(p);
        }
         
        // Function to find all result of factorial number
        static long factorialDivisors(int n)
        {
            sieve(n); // create sieve
         
            // Initialize result
            long result = 1;
         
            // find exponents of all primes which divides n
            // and less than n
            for (int i=0; i < allPrimes.size(); i++)
            {
                // Current divisor
                long p = allPrimes.get(i);
         
                // Find the highest power (stored in exp)'
                // of allPrimes[i] that divides n using
                // Legendre's formula.
                long exp = 0;
                while (p <= n)
                {
                    exp = exp + (n/p);
                    p = p*allPrimes.get(i);
                }
         
                // Multiply exponents of all primes less
                // than n
                result = result*(exp+1);
            }
         
            // return total divisors
            return result;
        }
         
        // Driver code
        public static void main(String []args)
        {
            System.out.println(factorialDivisors(6));
             
        }
 
 
}
 
//This code is contributed by ihritik

Python3




# Python3 program to find count
# 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
            i = p * 2;
            while(i <= n):
                prime[i] = False;
                i += p;
        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];
 
        # Multiply exponents of all
        # primes less than n
        result = result * (exp + 1);
 
    # return total divisors
    return result;
 
# Driver Code
print(factorialDivisors(6));
 
# This code is contributed by mits

C#




// C# program to find count 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];
         
        for(int i = 0; i <= n; i++)
        prime[i] = 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.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];
                }
         
                // Multiply exponents of all primes less
                // than n
                result = result * (exp + 1);
            }
         
            // return total divisors
            return result;
        }
         
        // Driver code
        public static void Main()
        {
            Console.WriteLine(factorialDivisors(6));
        }
}
 
//This code is contributed by chandan_jnu

PHP




<?php
// PHP program to find count 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];
        }
 
        // Multiply exponents of all primes
        // less than n
        $result = $result * ($exp + 1);
    }
 
    // return total divisors
    return $result;
}
 
// Driver Code
echo factorialDivisors(6);
 
// This code is contributed by mits
?>
Output
30

This article is contributed by Shashank Mishra ( Gullu ). This article is reviewed by team GeeksforGeeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

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