Maximum number of prime factors a number can have with exactly x factors

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Given an integer X, denoting the number of factors of a positive integer N can have. The task is to find the maximum number of distinct prime factors the number N can have.

Examples:

Input: X = 9
Output: 2
Explanation:
Some of the possible numbers having 9 factors are:
256: 1, 2, 4, 8, 16, 32, 64, 128, 256
Number of prime factors = 1
36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Number of prime factors = 2

Input: X = 8
Output: 3
Some of the numbers having 8 factors are:
128 : 1, 2, 4, 8, 16, 32, 64, 128
Number of prime factors = 1
24 : 1, 2, 3, 4, 6, 8, 12, 24
Number of prime factors = 2
30 : 1, 2, 3, 5, 6, 10, 15, 30
Number of prime factors = 3

Approach: The key observation in the problem is, any positive natural number can be represented as product of its prime factors as follows:

// Number can be represented as product
// prime factors as follows

// Total number of factors of N can be // defined as followsNumber of Factors = (p+1) * (q+1) * (r+1)..

In the above problem, the number of factors are given which can be used to find the maximum prime factors possible for a number with the given count of factors as follows:

If X can be expressed as product of K numbers then we have at most K primes in X.
In Order to split X as product of maximum number of values,
all the values should be prime.

X = (p+1) * (q+1) * (r+1)

// So the maximum number of prime
// factors of the given number greater
// than 1 can lead to a number N.
Let's say X = 12
X = 2 * 2 * 3
Then possible N can be:
N = a^(2-1) * b^(2-1) * c^(3-1)
N = a¹ * b¹ * c²

// Here a, b, and c can be any distinct prime
// numbers to get the possible value of N
N = 2¹ * 3¹ * 5²
N = 150

let's say X = 8
X = 2 * 2 * 2
N = 2¹ * 3¹ * 5¹
N  = 30

Therefore, the maximum count of prime divisors of a number can have is the count of the prime factors (can be repetitive also) in the factorization of the count of factors of the number.

Below is the implementation of the above approach:

C++

// C++ implementation to find the
// maximum count of the prime factors
// by the count of factors of number
 
#include <iostream>
#include <math.h>
 
using namespace std;
 
// Function to count number
// of prime factors of x
int countPrimeFactors(int n)
{
    if (n == 1)
        return 0;
 
    // Count variable is
    // incremented for every
    // prime factor of x
    int cnt = 0;
    while (n % 2 == 0) {
        cnt++;
        n = n / 2;
    }
 
    // Loop to count the number
    // of the prime factors of
    // the given number
    for (int i = 3; i <= sqrt(n);
         i += 2) {
        while (n % i == 0) {
            cnt++;
            n = n / i;
        }
    }
 
    if (n > 2)
        cnt++;
 
    return cnt;
}
 
// Driver Code
int main()
{
    int x = 8;
    int prime_factor_cnt = countPrimeFactors(x);
    cout << prime_factor_cnt << endl;
    return 0;
}

Java

// Java implementation to find the
// maximum count of the prime factors
// by the count of factors of number
import java.io.*;
public class GFG {
 
    // Function to count number
    // of prime factors of x
    static int countPrimeFactors(int n)
    {
        if (n == 1)
            return 0;
 
        // Count variable is
        // incremented form every
        // prime factor of x
        int cnt = 0;
        while (n % 2 == 0) {
            cnt++;
            n = n / 2;
        }
 
        // Loop to count the number
        // of the prime factors of
        // the given number
        for (int i = 3; i <= Math.sqrt(n); i += 2) {
            while (n % i == 0) {
                cnt++;
                n = n / i;
            }
        }
 
        if (n > 2)
            cnt++;
        return cnt;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int x = 8;
        int prime_factor_cnt = countPrimeFactors(x);
        System.out.print(prime_factor_cnt + "\n");
    }
}
 
// This code is contributed by Princi Singh

Python3

# Python3 implementation to find the
# maximum count of the prime factors
# by the count of factors of number
import math
 
# Function to count number 
# of prime factors of x
def countPrimeFactors(n):
     
    if (n == 1):
        return 0
         
    # Count variable is 
    # incremented form every 
    # prime factor of x
    cnt = 0
     
    while (n % 2 == 0):
        cnt += 1
        n = n // 2
         
    # Loop to count the number 
    # of the prime factors of
    # the given number
    for i in range(3, int(math.sqrt(n)) + 1, 2):
        while (n % i == 0):
            cnt += 1
            n = n // i
     
    if (n > 2):
        cnt += 1
     
    return cnt
 
# Driver Code
x = 8
prime_factor_cnt = countPrimeFactors(x)
 
print(prime_factor_cnt)
 
# This code is contributed by ShubhamCoder

C#

// C# implementation to find the
// maximum count of the prime factors
// by the count of factors of number
using System;
 
class GFG {
 
    // Function to count number
    // of prime factors of x
    static int countPrimeFactors(int n)
    {
        if (n == 1)
            return 0;
 
        // Count variable is
        // incremented form every
        // prime factor of x
        int cnt = 0;
        while (n % 2 == 0) {
            cnt++;
            n = n / 2;
        }
 
        // Loop to count the number
        // of the prime factors of
        // the given number
        for (int i = 3;
             i <= Math.Sqrt(n); i += 2) {
            while (n % i == 0) {
                cnt++;
                n = n / i;
            }
        }
 
        if (n > 2)
            cnt++;
 
        return cnt;
    }
 
    // Driver Code
    static public void Main()
    {
        int x = 8;
        int prime_factor_cnt = countPrimeFactors(x);
        Console.Write(prime_factor_cnt);
    }
}
 
// This code is contributed by ShubhamCoder

Javascript

<script>
 
// Javascript implementation to find the
// maximum count of the prime factors
// by the count of factors of number
 
// Function to count number
// of prime factors of x
function countPrimeFactors(n)
{
    if (n == 1)
        return 0;
 
    // Count variable is
    // incremented for every
    // prime factor of x
    let cnt = 0;
     
    while (n % 2 == 0)
    {
        cnt++;
        n = parseInt(n / 2);
    }
 
    // Loop to count the number
    // of the prime factors of
    // the given number
    for(let i = 3; i <= Math.sqrt(n); i += 2) 
    {
        while (n % i == 0) 
        {
            cnt++;
            n = parseInt(n / i);
        }
    }
 
    if (n > 2)
        cnt++;
 
    return cnt;
}
 
// Driver Code
let x = 8;
let prime_factor_cnt = countPrimeFactors(x);
 
document.write(prime_factor_cnt);
 
// This code is contributed by souravmahato348
 
</script>

Output:

Time Complexity: O(N^1/2)

Auxiliary Space: O(1)

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