Check if a number has prime count of divisors

Given an integer N, the task is to check if the count of divisors of N is prime or not.

Examples:

Input: N = 13 
Output: Yes 
The divisor count is 2 (1 and 13) which is prime.

Input: N = 8 
Output: No 
The divisors are 1, 2, 4 and 8. 

Approach: Please read this article to find the count of divisors of a number. So find the maximum value of i for every prime divisor p such that N % (pⁱ) = 0. So the count of divisors gets multiplied by (i + 1). The count of divisors will be (i¹ + 1) * (i² + 1) * … * (i^k + 1). 
It can now be seen that there can only be one prime divisor for the maximum i and if N % pⁱ = 0 then (i + 1) should be prime. The primality can be checked in sqrt(n) time and the prime factors can also be found in sqrt(n) time. So the overall time complexity will be O(sqrt(n)).

Below is the implementation of the above approach:  

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;

// Function that returns true
// if n is prime
bool Prime(int n)
{
    // There is no prime
    // less than 2
    if (n < 2)
        return false;

    // Run a loop from 2 to sqrt(n)
    for (int i = 2; i <= sqrt(n); i++)

        // If there is any factor
        if (n % i == 0)
            return false;

    return true;
}

// Function that returns true if n
// has a prime count of divisors
bool primeCountDivisors(int n)
{
    if (n < 2)
        return false;

    // Find the prime factors
    for (int i = 2; i <= sqrt(n); i++)
        if (n % i == 0) {

            // Find the maximum value of i for every
            // prime divisor p such that n % (p^i) == 0
            long a = n, c = 0;
            while (a % i == 0) {
                a /= i;
                c++;
            }

            // If c+1 is a prime number and a = 1
            if (a == 1 && Prime(c + 1))
                return true;

            // The number cannot have two factors
            // to have count of divisors prime
            else
                return false;
        }

    // Else the number is prime so
    // it has only two divisors
    return true;
}

// Driver code
int main()
{
    int n = 13;

    if (primeCountDivisors(n))
        cout << "Yes";
    else
        cout << "No";

    return 0;
}

Java

// Java implementation of the approach
class GFG
{

    // Function that returns true
    // if n is prime
    static boolean Prime(int n)
    {
        // There is no prime
        // less than 2
        if (n < 2)
            return false;

        // Run a loop from 2 to sqrt(n)
        for (int i = 2; i <= (int)Math.sqrt(n); i++)

            // If there is any factor
            if (n % i == 0)
                return false;
        return true;
    }

    // Function that returns true if n
    // has a prime count of divisors
    static boolean primeCountDivisors(int n)
    {
        if (n < 2)
            return false;

        // Find the prime factors
        for (int i = 2; i <= (int)Math.sqrt(n); i++)
            if (n % i == 0)
            {

                // Find the maximum value of i for every
                // prime divisor p such that n % (p^i) == 0
                long a = n, c = 0;
                while (a % i == 0)
                {
                    a /= i;
                    c++;
                }

                // If c+1 is a prime number and a = 1
                if (a == 1 && Prime((int)c + 1) == true)
                    return true;

                // The number cannot have two factors
                // to have count of divisors prime
                else
                    return false;
            }

        // Else the number is prime so
        // it has only two divisors
        return true;
    }

    // Driver code
    public static void main (String[] args)
    {
        int n = 13;

        if (primeCountDivisors(n))
            System.out.println("Yes");
        else
            System.out.println("No");
    }
}

// This code is contributed by AnkitRai01

Python3

# Python3 implementation of the approach
from math import sqrt

# Function that returns true
# if n is prime
def Prime(n) :

    # There is no prime
    # less than 2
    if (n < 2) :
        return False;

    # Run a loop from 2 to sqrt(n)
    for i in range(2, int(sqrt(n)) + 1) :

        # If there is any factor
        if (n % i == 0) :
            return False;

    return True;

# Function that returns true if n
# has a prime count of divisors
def primeCountDivisors(n) :

    if (n < 2) :
        return False;

    # Find the prime factors
    for i in range(2, int(sqrt(n)) + 1) :
        if (n % i == 0) :

            # Find the maximum value of i for every
            # prime divisor p such that n % (p^i) == 0
            a = n; c = 0;
            while (a % i == 0) :
                a //= i;
                c += 1;

            # If c + 1 is a prime number and a = 1
            if (a == 1 and Prime(c + 1)) :
                return True;

            # The number cannot have two factors
            # to have count of divisors prime
            else :
                return False;

    # Else the number is prime so
    # it has only two divisors
    return True;

# Driver code
if __name__ == "__main__" :

    n = 13;

    if (primeCountDivisors(n)) :
        print("Yes");
    else :
        print("No");

# This code is contributed by AnkitRai01

C#

// C# implementation of the approach
using System;

class GFG
{

    // Function that returns true
    // if n is prime
    static bool Prime(int n)
    {

        // There is no prime
        // less than 2
        if (n < 2)
            return false;

        // Run a loop from 2 to sqrt(n)
        for (int i = 2; i <= (int)Math.Sqrt(n); i++)

            // If there is any factor
            if (n % i == 0)
                return false;
        return true;
    }

    // Function that returns true if n
    // has a prime count of divisors
    static bool primeCountDivisors(int n)
    {
        if (n < 2)
            return false;

        // Find the prime factors
        for (int i = 2; i <= (int)Math.Sqrt(n); i++)
            if (n % i == 0)
            {

                // Find the maximum value of i for every
                // prime divisor p such that n % (p^i) == 0
                long a = n, c = 0;
                while (a % i == 0)
                {
                    a /= i;
                    c++;
                }

                // If c+1 is a prime number and a = 1
                if (a == 1 && Prime((int)c + 1) == true)
                    return true;

                // The number cannot have two factors
                // to have count of divisors prime
                else
                    return false;
            }

        // Else the number is prime so
        // it has only two divisors
        return true;
    }

    // Driver code
    public static void Main()
    {
        int n = 13;

        if (primeCountDivisors(n))
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No");
    }
}

// This code is contributed by AnkitRai01

Javascript

<script>

// Javascript implementation of the approach

// Function that returns true
// if n is prime
function Prime(n)
{

    // There is no prime
    // less than 2
    if (n < 2)
        return false;

    // Run a loop from 2 to sqrt(n)
    for(var i = 2; i <= Math.sqrt(n); i++)

        // If there is any factor
        if (n % i == 0)
            return false;

    return true;
}

// Function that returns true if n
// has a prime count of divisors
function primeCountDivisors( n)
{
    if (n < 2)
        return false;

    // Find the prime factors
    for(var i = 2; i <= Math.sqrt(n); i++)
        if (n % i == 0)
        {

            // Find the maximum value of i for every
            // prime divisor p such that n % (p^i) == 0
            var a = n, c = 0;

            while (a % i == 0)
            {
                a /= i;
                c++;
            }

            // If c+1 is a prime number and a = 1
            if (a == 1 && Prime(c + 1))
                return true;

            // The number cannot have two factors
            // to have count of divisors prime
            else
                return false;
        }

    // Else the number is prime so
    // it has only two divisors
    return true;
}

// Driver rcode
n = 13;

if (primeCountDivisors(n))
    document.write("Yes");
else
    document.write("No");

// This code is contributed by SoumikMondal

</script>

Output

Yes

Time Complexity: O(sqrt(n)), as we are using a loop to traverse sqrt (n) times. Where n is the integer given as input.

Auxiliary Space: O(1), as we are not using any extra space.

Approach 2: HashMap:

• The approach uses a map to count the frequency of each prime factor. A map is a data structure in C++ that stores key-value pairs. In this case, the key is the prime factor, and the value is the frequency of that prime factor. The map is used because it allows us to keep track of the frequency of each prime factor efficiently without needing to know the prime factors beforehand.
• The approach then uses a for loop to iterate over all the prime factors and count their frequency using the map. It also checks if there is any remaining factor greater than 1 after dividing by all the prime factors up to the square root of the number. If there is such a factor, it is also included in the map with a frequency of 1.
• After counting the frequency of each prime factor, the approach uses another for loop to calculate the total number of divisors of the number using the formula (f1+1) * (f2+1) * … * (fn+1), where f1, f2, …, fn are the frequencies of each prime factor in the factorization of the number. This formula works because for each prime factor, we have (frequency + 1) choices for how many times to include that prime factor in a divisor. We multiply all these choices together to get the total number of divisors.
• Finally, the approach checks if the total number of divisors is a prime number using another for loop that checks if the number is divisible by any number from 2 up to the square root of the number of divisors. If it is divisible by any of these numbers, then it is not a prime number and the function returns false. If none of these numbers divide the number of divisors, then it is a prime number, and the function returns true.

Here is the code of this approach:

C++

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

// Function that returns true if n
// has a prime count of divisors
bool primeCountDivisors(int n)
{
    if (n < 2)
        return false;

    // Find all the prime factors of n
    vector<int> primes;
    for (int i = 2; i <= sqrt(n); i++) {
        while (n % i == 0) {
            primes.push_back(i);
            n /= i;
        }
    }
    if (n > 1) {
        primes.push_back(n);
    }

    // Count the frequency of each prime factor
    map<int, int> freq;
    for (int prime : primes) {
        freq[prime]++;
    }

    // Calculate the number of divisors of n
    int numDivisors = 1;
    for (auto it : freq) {
        numDivisors *= (it.second + 1);
    }

    // Check if the number of divisors is a prime number or not
    if (numDivisors < 2)
        return false;
    for (int i = 2; i <= sqrt(numDivisors); i++) {
        if (numDivisors % i == 0)
            return false;
    }
    return true;
}

// Driver code
int main()
{
    int n = 13;

    if (primeCountDivisors(n))
        cout << "Yes";
    else
        cout << "No";

    return 0;
}

Java

import java.util.HashMap;
import java.util.Map;

public class PrimeCountDivisors {

    public static boolean primeCountDivisors(int n) {
        if (n < 2) {
            return false;
        }

        // Find all the prime factors of n and their occurrences using a HashMap
        Map<Integer, Integer> primes = new HashMap<>();
        for (int i = 2; i <= Math.sqrt(n); i++) {
            while (n % i == 0) {
                // Increment the count of the prime factor in the HashMap
                primes.put(i, primes.getOrDefault(i, 0) + 1);
                n /= i; // Reduce n by dividing it by the prime factor
            }
        }

        // If n is still greater than 1, it is a prime factor itself
        if (n > 1) {
            primes.put(n, 1);
        }

        // Calculate the number of divisors of n using the prime factorization
        int numDivisors = 1;
        for (Map.Entry<Integer, Integer> entry : primes.entrySet()) {
            numDivisors *= (entry.getValue() + 1);
        }

        // Check if the number of divisors is a prime number or not
        // If the number of divisors is less than 2, it is not prime
        if (numDivisors < 2) {
            return false;
        }

        // Check if the number of divisors is prime by checking for divisors from 2 to sqrt(numDivisors)
        for (int i = 2; i <= Math.sqrt(numDivisors); i++) {
            if (numDivisors % i == 0) {
                return false; // If a divisor is found, numDivisors is not a prime number
            }
        }

        // If no divisors are found other than 1 and numDivisors, it is a prime number
        return true;
    }

    public static void main(String[] args) {
        int n = 13;

        // Check if the number of divisors of 'n' is a prime number
        if (primeCountDivisors(n)) {
            System.out.println("Yes"); // If it's prime, print "Yes"
        } else {
            System.out.println("No"); // If it's not prime, print "No"
        }
    }
}

Python3

import math

# Function that returns True if n has a prime count of divisors
def primeCountDivisors(n):
    if n < 2:
        return False

    # Find all the prime factors of n
    primes = []
    i = 2
    while i <= math.sqrt(n):
        while n % i == 0:
            primes.append(i)
            n //= i
        i += 1
    if n > 1:
        primes.append(n)

    # Count the frequency of each prime factor
    freq = {}
    for prime in primes:
        freq[prime] = freq.get(prime, 0) + 1

    # Calculate the number of divisors of n
    numDivisors = 1
    for count in freq.values():
        numDivisors *= (count + 1)

    # Check if the number of divisors is a prime number or not
    if numDivisors < 2:
        return False
    for i in range(2, int(math.sqrt(numDivisors)) + 1):
        if numDivisors % i == 0:
            return False
    return True

# Driver code
if __name__ == "__main__":
    n = 13
    if primeCountDivisors(n):
        print("Yes")
    else:
        print("No")

C#

using System;
using System.Collections.Generic;

class Program {
    static bool PrimeCountDivisors(int n)
    {
        if (n < 2)
            return false;

        // Find all the prime factors of n
        List<int> primes = new List<int>();
        for (int i = 2; i <= Math.Sqrt(n); i++) {
            while (n % i == 0) {
                primes.Add(i);
                n /= i;
            }
        }
        if (n > 1) {
            primes.Add(n);
        }

        // Count the frequency of each prime factor
        Dictionary<int, int> freq
            = new Dictionary<int, int>();
        foreach(int prime in primes)
        {
            if (freq.ContainsKey(prime)) {
                freq[prime]++;
            }
            else {
                freq[prime] = 1;
            }
        }

        // Calculate the number of divisors of n
        int numDivisors = 1;
        foreach(KeyValuePair<int, int> pair in freq)
        {
            numDivisors *= (pair.Value + 1);
        }

        // Check if the number of divisors is a prime number
        // or not
        if (numDivisors < 2)
            return false;
        for (int i = 2; i <= Math.Sqrt(numDivisors); i++) {
            if (numDivisors % i == 0)
                return false;
        }
        return true;
    }

    static void Main()
    {
        int n = 13;

        if (PrimeCountDivisors(n))
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No");
    }
}
// This code is contributed by sarojmcy2e

Javascript

// Function that returns true if n
// has a prime count of divisors
function primeCountDivisors(n) {
    if (n < 2) return false;

    // Find all the prime factors of n
    let primes = [];
    for (let i = 2; i <= Math.sqrt(n); i++) {
        while (n % i === 0) {
            primes.push(i);
            n /= i;
        }
    }
    if (n > 1) {
        primes.push(n);
    }

    // Count the frequency of each prime factor
    let freq = {};
    for (let prime of primes) {
        freq[prime] = (freq[prime] || 0) + 1;
    }

    // Calculate the number of divisors of n
    let numDivisors = 1;
    for (let key in freq) {
        numDivisors *= (freq[key] + 1);
    }

    // Check if the number of divisors is a prime number or not
    if (numDivisors < 2) return false;
    for (let i = 2; i <= Math.sqrt(numDivisors); i++) {
        if (numDivisors % i === 0) return false;
    }
    return true;
}

// Driver code
let n = 13;

if (primeCountDivisors(n)) {
    console.log("Yes");
} else {
    console.log("No");
}

Output: 

Yes

Time Complexity: O(sqrt(N)), as we are using a loop to traverse sqrt (N) times. Where n is the integer given as input.

Auxiliary Space: O(LogN), as we are not using any extra space.