Count of the non-prime divisors of a given number

Given a number N, the task is to find the count of non-prime divisors of the given number N.

Examples:

Input: N = 8 
Output: 3 
Explanation:
Divisors of 8 are – {1, 2, 4, 8} 
Non-Prime Divisors – {1, 4, 8}

Input: N = 20 
Output: 4 
Explanation:
Divisors of 20 are – {1, 2, 4, 5, 10, 20} 
Non-Prime Divisors – {1, 4, 10, 20} 

Approach: The key observation in the problem is that any number can be written as a product of its prime factors as 

where K is the count of the prime factors of the given number. Using permutation and combination we can find that the total count of the 
factors that are 

Therefore, the count of the non-prime factors will be: 

Below is the implementation of the above approach: 

C++

// C++ program to find count of
// non-prime divisors of given number

#include <bits/stdc++.h>
using namespace std;

// Function to factors of the given
// number
vector<int> getFactorization(int x)
{
    int count = 0;
    vector<int> v;

    // Loop to find the divisors of
    // the number 2
    while (x % 2 == 0) {
        count++;
        x = x / 2;
    }
    if (count != 0)
        v.push_back(count);

    // Loop to find the divisors of the
    // given number upto SQRT(N)
    for (int i = 3; i <= sqrt(x); i += 2) {
        count = 0;
        while (x % i == 0) {
            count++;
            x /= i;
        }
        if (count != 0)
            v.push_back(count);
    }

    // Condition to check if the rest
    // number is also a prime number
    if (x > 1) {
        v.push_back(1);
    }
    return v;
}

// Function to find the non-prime
// divisors of the given number
int nonPrimeDivisors(int N)
{
    vector<int> v = getFactorization(N);
    int ret = 1;

    // Loop to count the number of
    // the total divisors of given number
    for (int i = 0; i < v.size(); i++)
        ret = ret * (v[i] + 1);

    ret = ret - v.size();
    return ret;
}

// Driver Code
int main()
{
    int N = 8;

    // Function Call
    cout << nonPrimeDivisors(N) << endl;
    return 0;
}

Java

// Java program to find
// count of non-prime
// divisors of given number
import java.util.*;
class GFG{

// Function to factors
// of the given number
static Vector<Integer> getFactorization(int x)
{
  int count = 0;
  Vector<Integer> v = new Vector<>();

  // Loop to find the
  // divisors of the number 2
  while (x % 2 == 0)
  {
    count++;
    x = x / 2;
  }

  if (count != 0)
    v.add(count);

  // Loop to find the divisors
  // of the given number upto SQRT(N)
  for (int i = 3;
           i <= Math.sqrt(x); i += 2)
  {
    count = 0;
    while (x % i == 0)
    {
      count++;
      x /= i;
    }

    if (count != 0)
      v.add(count);
  }

  // Condition to check if
  // the rest number is also
  // a prime number
  if (x > 1)
  {
    v.add(1);
  }
  return v;
}

// Function to find the non-prime
// divisors of the given number
static int nonPrimeDivisors(int N)
{
  Vector<Integer> v = getFactorization(N);
  int ret = 1;

  // Loop to count the number of
  // the total divisors of given number
  for (int i = 0; i < v.size(); i++)
    ret = ret * (v.get(i) + 1);

  ret = ret - v.size();
  return ret;
}

// Driver Code
public static void main(String[] args)
{
  int N = 8;

  // Function Call
  System.out.println(nonPrimeDivisors(N));
}
}

// This code is contributed by shikhasingrajput

Python3

# Python3 program to find count of
# non-prime divisors of given number
from math import sqrt

# Function to factors of the given
# number
def getFactorization(x):

    count = 0
    v = []

    # Loop to find the divisors of
    # the number 2
    while (x % 2 == 0):
        count += 1
        x = x // 2

    if (count != 0):
        v.append(count)

    # Loop to find the divisors of the
    # given number upto SQRT(N)
    for i in range(3, int(sqrt(x)) + 12):
        count = 0

        while (x % i == 0):
            count += 1
            x //= i

        if (count != 0):
            v.append(count)

    # Condition to check if the rest
    # number is also a prime number
    if (x > 1):
        v.append(1)

    return v

# Function to find the non-prime
# divisors of the given number
def nonPrimeDivisors(N):

    v = getFactorization(N)
    ret = 1

    # Loop to count the number of
    # the total divisors of given number
    for i in range(len(v)):
        ret = ret * (v[i] + 1)
    ret = ret - len(v)

    return ret

# Driver Code
if __name__ == '__main__':

    N = 8

    # Function Call
    print(nonPrimeDivisors(N))

# This code is contributed by Samarth

C#

// C# program to find
// count of non-prime
// divisors of given number
using System;
using System.Collections.Generic;
class GFG{

// Function to factors
// of the given number
static List<int> getFactorization(int x)
{
  int count = 0;
  List<int> v = new List<int>();

  // Loop to find the
  // divisors of the number 2
  while (x % 2 == 0)
  {
    count++;
    x = x / 2;
  }

  if (count != 0)
    v.Add(count);

  // Loop to find the divisors
  // of the given number upto
  // SQRT(N)
  for (int i = 3;
           i <= Math.Sqrt(x); i += 2)
  {
    count = 0;
    while (x % i == 0)
    {
      count++;
      x /= i;
    }

    if (count != 0)
      v.Add(count);
  }

  // Condition to check if
  // the rest number is also
  // a prime number
  if (x > 1)
  {
    v.Add(1);
  }
  return v;
}

// Function to find the non-prime
// divisors of the given number
static int nonPrimeDivisors(int N)
{
  List<int> v = getFactorization(N);
  int ret = 1;

  // Loop to count the number of
  // the total divisors of given number
  for (int i = 0; i < v.Count; i++)
    ret = ret * (v[i] + 1);

  ret = ret - v.Count;
  return ret;
}

// Driver Code
public static void Main(String[] args)
{
  int N = 8;

  // Function Call
  Console.WriteLine(nonPrimeDivisors(N));
}
}

// This code is contributed by gauravrajput1

Javascript

<script>

// Javascript program to find
// count of non-prime
// divisors of given number

// Function to factors
// of the given number
function getFactorization(x)
{
      let count = 0;
      let v = [];

      // Loop to find the
      // divisors of the number 2
      while (x % 2 == 0)
      {
        count++;
        x = Math.floor(x / 2);
      }

      if (count != 0)
        v.push(count);

      // Loop to find the divisors
      // of the given number upto
      // SQRT(N)
      for (let i = 3;
        i <= Math.floor(Math.sqrt(x)); i += 2)
      {
        count = 0;
        while (x % i == 0)
        {
              count++;
              x = Math.floor(x / i);
        }

        if (count != 0)
          v.push(count);
      }

      // Condition to check if
      // the rest number is also
      // a prime number
      if (x > 1)
      {
        v.push(1);
      }
      return v;
}

// Function to find the non-prime
// divisors of the given number
function nonPrimeDivisors(N)
{
      let v = getFactorization(N);
      let ret = 1;

      // Loop to count the number of
      // the total divisors of given number
      for (let i = 0; i < v.length; i++)
        ret = ret * (v[i] + 1);

      ret = ret - v.length;
      return ret;
}

// Driver Code

    let N = 8;

  // Function Call
  document.write(nonPrimeDivisors(N));

</script>

Output:

3