Compositorial of a number

Given a natural number N, the task is to find the Nth compositorial number.

Compositorial of a number refers to the product of all the positive composite integers up to N.
The compositorial of a number N is denoted by \frac{N!}{N#} where N! is the factorial of the number and N# is the Primorial of the number N.

Examples:

Input: N = 4
Output: 1728
Explanation:
The first 4 composite numbers are 4, 6, 8, 9. Therefore, the compositorial is the product of all of the numbers.

Input: N = 5
Output: 17280



Approach: The following steps can be followed to compute the Nth compositorial number.

  1. Get the number N.
  2. Find all the composite numbers upto N.
  3. Product the obtained composite numbers.
  4. Print the product.

Below is the implementation of the above approach:

C++

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// C++ program to find compositorial
// of composite numbers
#include <bits/stdc++.h>
using namespace std;
  
vector<int> compo;
  
// Function to check if
// a number is composite.
bool isComposite(int n)
{
      
    // Corner cases
    if (n <= 3)
        return false;
  
    // This is checked so that we can
    // skip the middle five numbers
    // in the below loop
    if (n % 2 == 0 or n % 3 == 0)
        return true;
  
    int i = 5;
    while(i * i <= n)
    {
        if (n % i == 0 or
            n % (i + 2) == 0)
            return true;
        i = i + 6;
    }
    return false;
}
  
// This function stores all
// composite numbers less than N
void Compositorial_list(int n)
{
    int l = 0;
    for(int i = 4; i < 1000000; i++)
    {
       if (l < n)
       {
           if (isComposite(i))
           {
               compo.push_back(i);
               l += 1;
           }
       }
    }
}
  
// Function to calculate 
// the compositorial of n
int calculateCompositorial(int n)
{
      
    // Multiply first n composite number
    int result = 1;
      
    for(int i = 0; i < n; i++)
        result = result * compo[i];
    return result;
}
  
// Driver code
int main()
{
    int n = 5;
      
    // Vector to store all the
    // composite less than N
    Compositorial_list(n);
      
    cout << (calculateCompositorial(n));
      
    return 0;
}
  
// This code is contributed by mohit kumar 29

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Python3

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# Python3 program to find Compositorial 
# of composite numbers  
   
# Function to check 
# if a number is composite. 
def isComposite(n): 
       
    # Corner cases 
    if (n <= 3): 
        return False
     
    # This is checked so that we can 
    # skip the middle five numbers 
    # in the below loop 
    if (n % 2 == 0 or n % 3 == 0): 
        return True
  
    i = 5
    while(i * i <= n): 
             
        if (n % i == 0\
            or n % (i + 2) == 0): 
            return True
        i = i + 6
             
    return False
       
# This function stores all  
# Composite numbers less than N
def Compositorial_list(n):
    l = 0
    for i in range(4, 10**6):
        if l<n:
            if isComposite(i):
                compo.append(i)
                l+= 1
           
     
# Function to calculate the 
# Compositorial of n  
def calculateCompositorial(n):
       
    # Multiply first n composite number  
    result = 1 
    for i in range(n): 
        result = result * compo[i]  
    return result  
     
# Driver code  
if __name__ == "__main__":
    n = 5
   
    # Vector to store all the 
    # composite less than N
    compo =[]
   
    Compositorial_list(n)
   
    print(calculateCompositorial(n))

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Output:

17280

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Improved By : mohit kumar 29