Count of all subsequence whose product is a Composite number

Given an array arr[], the task is to find the number of non-empty subsequences from the given array such that the product of subsequence is a composite number.

Example:

Input: arr[] = {2, 3, 4}
Output: 5
Explanation:
There are 5 subsequences whose product is composite number {4}, {2, 3}, {2, 4}, {3, 4}, {2, 3, 4}.

Input: arr[] = {2, 1, 2}
Output: 2
Explanation:
There is 2 subsequences whose product is composite number {2, 2}, {2, 1, 2}

Approach: The approach used to find the count of such subsequences is similar to the approach used in this article. Also, the approach can slightly tweaked to get the count of subsequences whose product is a Prime number.



To solve the problem mentioned above, we have to find the total number of non-empty subsequences and subtract the subsequence whose product is not a composite number. The 3 possible cases where the product is not a composite number are:

  • Any nonempty combination of 1 that is

    pow(2, count of “1”) – 1

  • Any subsequence of length 1 which consists of a prime number that is basically the

    count of prime numbers

  • Combination of non-empty 1 with a prime number that is

    (pow(2, number of 1 ) – 1) * (count of prime numbers)

Below is the implementation of above approach:

C++

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// C++ implementation to count all
// subsequence whose product
// is Composite number
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to check whether a
// number is prime or not
bool isPrime(int n)
{
    if (n <= 1)
        return false;
    for (int i = 2; i < n; i++)
        if (n % i == 0)
  
            return false;
  
    return true;
}
  
// Function to find number of subsequences
// whose product is a composite number
int countSubsequences(int arr[], int n)
{
    // Find total non empty subsequence
    int totalSubsequence = pow(2, n) - 1;
  
    int countPrime = 0, countOnes = 0;
  
    // Find count of prime number and ones
    for (int i = 0; i < n; i++) {
        if (arr[i] == 1)
            countOnes++;
        else if (isPrime(arr[i]))
            countPrime++;
    }
  
    int compositeSubsequence;
  
    // Calculate the non empty one subsequence
    int onesSequence = pow(2, countOnes) - 1;
  
    // Find count of composite subsequence
    compositeSubsequence
        = totalSubsequence - countPrime
          - onesSequence
          - onesSequence * countPrime;
  
    return compositeSubsequence;
}
  
// Driver code
int main()
{
  
    int arr[] = { 2, 1, 2 };
  
    int n = sizeof(arr) / sizeof(arr[0]);
  
    cout << countSubsequences(arr, n);
  
    return 0;
}

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Java

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// Java implementation to count all
// subsequence whose product
// is Composite number
import java.util.*;
class GFG{
  
// Function to check whether a
// number is prime or not
static boolean isPrime(int n)
{
    if (n <= 1)
        return false;
    for (int i = 2; i < n; i++)
        if (n % i == 0)
  
            return false;
  
    return true;
}
  
// Function to find number of subsequences
// whose product is a composite number
static int countSubsequences(int arr[], int n)
{
    // Find total non empty subsequence
    int totalSubsequence = (int)(Math.pow(2, n) - 1);
  
    int countPrime = 0, countOnes = 0;
  
    // Find count of prime number and ones
    for (int i = 0; i < n; i++) 
    {
        if (arr[i] == 1)
            countOnes++;
        else if (isPrime(arr[i]))
            countPrime++;
    }
  
    int compositeSubsequence;
  
    // Calculate the non empty one subsequence
    int onesSequence = (int)(Math.pow(2, countOnes) - 1);
  
    // Find count of composite subsequence
    compositeSubsequence = totalSubsequence - 
                                 countPrime -
                               onesSequence -
                               onesSequence * 
                               countPrime;
  
    return compositeSubsequence;
}
  
// Driver code
public static void main(String[] args)
{
    int arr[] = { 2, 1, 2 };
  
    int n = arr.length;
  
    System.out.print(countSubsequences(arr, n));
}
}
  
// This code is contributed by 29AjayKumar

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Python3

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# Python3 implementation to count 
# all subsequence whose product
# is composite number
  
# Function to check whether 
# a number is prime or not
def isPrime(n):
      
    if (n <= 1):
        return False;
          
    for i in range(2, n):
        if (n % i == 0):
            return False;
  
    return True;
  
# Function to find number of subsequences
# whose product is a composite number
def countSubsequences(arr, n):
      
    # Find total non empty subsequence
    totalSubsequence = (int)(pow(2, n) - 1);
      
    countPrime = 0;
    countOnes = 0;
  
    # Find count of prime number and ones
    for i in range(n):
        if (arr[i] == 1):
            countOnes += 1;
              
        elif (isPrime(arr[i])):
            countPrime += 1;
  
    compositeSubsequence = 0;
  
    # Calculate the non empty one subsequence
    onesSequence = (int)(pow(2, countOnes) - 1);
  
    # Find count of composite subsequence
    compositeSubsequence = (totalSubsequence - 
                                  countPrime - 
                                onesSequence - 
                                onesSequence * 
                                  countPrime);
  
    return compositeSubsequence;
  
# Driver code
if __name__ == '__main__':
      
    arr = [ 2, 1, 2 ];
    n = len(arr);
  
    print(countSubsequences(arr, n));
  
# This code is contributed by 29AjayKumar

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C#

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// C# implementation to count all
// subsequence whose product
// is Composite number
using System;
class GFG{
  
// Function to check whether a
// number is prime or not
static bool isPrime(int n)
{
    if (n <= 1)
        return false;
    for (int i = 2; i < n; i++)
        if (n % i == 0)
  
            return false;
  
    return true;
}
  
// Function to find number of subsequences
// whose product is a composite number
static int countSubsequences(int []arr, int n)
{
    // Find total non empty subsequence
    int totalSubsequence = (int)(Math.Pow(2, n) - 1);
  
    int countPrime = 0, countOnes = 0;
  
    // Find count of prime number and ones
    for (int i = 0; i < n; i++) 
    {
        if (arr[i] == 1)
            countOnes++;
        else if (isPrime(arr[i]))
            countPrime++;
    }
  
    int compositeSubsequence;
  
    // Calculate the non empty one subsequence
    int onesSequence = (int)(Math.Pow(2, countOnes) - 1);
  
    // Find count of composite subsequence
    compositeSubsequence = totalSubsequence - 
                                 countPrime -
                               onesSequence -
                               onesSequence * 
                                 countPrime;
  
    return compositeSubsequence;
}
  
// Driver code
public static void Main()
{
    int []arr = { 2, 1, 2 };
  
    int n = arr.Length;
  
    Console.Write(countSubsequences(arr, n));
}
}
  
// This code is contributed by Nidhi_biet

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Output:

2

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Improved By : 29AjayKumar, nidhi_biet