Print all prime factors and their powers

Given a number N, print all its unique prime factors and their powers in N. 

Examples:

Input: N = 100
Output: Factor Power
          2      2
          5      2

Input: N = 35
Output: Factor  Power
          5      1
          7      1

A Simple Solution is to first find prime factors of N. Then for every prime factor, find the highest power of it that divides N and print it.
An Efficient Solution is to use Sieve of Eratosthenes. 

1) First compute an array s[N+1] using Sieve of Eratosthenes.

s[i] = Smallest prime factor of "i" that
       divides "i".

For example let N  = 10
  s[2] = s[4] = s[6] = s[8] = s[10] = 2;
  s[3] = s[9] = 3;
  s[5] = 5;
  s[7] = 7;


2) Using the above computed array s[],
   we can find all powers in O(Log N) time.

    curr = s[N];  // Current prime factor of N
    cnt = 1;   // Power of current prime factor

    // Printing prime factors and their powers
    while (N > 1)
    {
        N /= s[N];

        // N is now N/s[N].  If new N also has its 
        // smallest prime factor as curr, increment 
        // power and continue
        if (curr == s[N])
        {
            cnt++;
            continue;
        }

        // Print prime factor and its power
        print(curr, cnt);

        // Update current prime factor as s[N] and
        // initializing count as 1.
        curr = s[N];
        cnt = 1;
    }

Below is the implementation of above steps.

C++

// C++ Program to print prime factors and their
// powers using Sieve Of Eratosthenes
#include<bits/stdc++.h>
using namespace std;

// Using SieveOfEratosthenes to find smallest prime
// factor of all the numbers.
// For example, if N is 10,
// s[2] = s[4] = s[6] = s[10] = 2
// s[3] = s[9] = 3
// s[5] = 5
// s[7] = 7
void sieveOfEratosthenes(int N, int s[])
{
    // Create a boolean array "prime[0..n]" and
    // initialize all entries in it as false.
    vector <bool> prime(N+1, false);

    // Initializing smallest factor equal to 2
    // for all the even numbers
    for (int i=2; i<=N; i+=2)
        s[i] = 2;

    // For odd numbers less than equal to n
    for (int i=3; i<=N; i+=2)
    {
        if (prime[i] == false)
        {
            // s(i) for a prime is the number itself
            s[i] = i;

            // For all multiples of current prime number
            for (int j=i; j*i<=N; j+=2)
            {
                if (prime[i*j] == false)
                {
                    prime[i*j] = true;

                    // i is the smallest prime factor for
                    // number "i*j".
                    s[i*j] = i;
                }
            }
        }
    }
}

// Function to generate prime factors and its power
void generatePrimeFactors(int N)
{
    // s[i] is going to store smallest prime factor
    // of i.
    int s[N+1];

    // Filling values in s[] using sieve
    sieveOfEratosthenes(N, s);

    printf("Factor Power\n");

    int curr = s[N];  // Current prime factor of N
    int cnt = 1;   // Power of current prime factor

    // Printing prime factors and their powers
    while (N > 1)
    {
        N /= s[N];

        // N is now N/s[N].  If new N also has smallest
        // prime factor as curr, increment power
        if (curr == s[N])
        {
            cnt++;
            continue;
        }

        printf("%d\t%d\n", curr, cnt);

        // Update current prime factor as s[N] and
        // initializing count as 1.
        curr = s[N];
        cnt = 1;
    }
}

//Driver Program
int main()
{
    int N = 360;
    generatePrimeFactors(N);
    return 0;
}

Java

// Java Program to print prime
// factors and their powers using
// Sieve Of Eratosthenes
import java.io.*;
public class GFG
{
// Using SieveOfEratosthenes
// to find smallest prime
// factor of all the numbers.
// For example, if N is 10,
// s[2] = s[4] = s[6] = s[10] = 2
// s[3] = s[9] = 3
// s[5] = 5
// s[7] = 7
static void sieveOfEratosthenes(int N,
                                int s[])
{
    // Create a boolean array
    // "prime[0..n]"  and initialize
    // all entries in it as false.
    boolean[] prime = new boolean[N + 1];

    // Initializing smallest
    // factor equal to 2
    // for all the even numbers
    for (int i = 2; i <= N; i += 2)
        s[i] = 2;

    // For odd numbers less
    // then equal to n
    for (int i = 3; i <= N; i += 2)
    {
        if (prime[i] == false)
        {
            // s(i) for a prime is
            // the number itself
            s[i] = i;

            // For all multiples of
            // current prime number
            for (int j = i; j * i <= N; j += 2)
            {
                if (prime[i * j] == false)
                {
                    prime[i * j] = true;

                    // i is the smallest prime
                    // factor for number "i*j".
                    s[i * j] = i;
                }
            }
        }
    }
}

// Function to generate prime
// factors and its power
static void generatePrimeFactors(int N)
{
    // s[i] is going to store
    // smallest prime factor of i.
    int[] s = new int[N + 1];

    // Filling values in s[] using sieve
    sieveOfEratosthenes(N, s);

    System.out.println("Factor Power");

    int curr = s[N]; // Current prime factor of N
    int cnt = 1; // Power of current prime factor

    // Printing prime factors
    // and their powers
    while (N > 1)
    {
        N /= s[N];

        // N is now N/s[N]. If new N
        // also has smallest prime
        // factor as curr, increment power
        if (curr == s[N])
        {
            cnt++;
            continue;
        }

        System.out.println(curr + "\t" + cnt);

        // Update current prime factor
        // as s[N] and initializing
        // count as 1.
        curr = s[N];
        cnt = 1;
    }
}

// Driver Code
public static void main(String[] args)
{
    int N = 360;
    generatePrimeFactors(N);
}
}

// This code is contributed by mits

Python3

# Python3 program to print prime
# factors and their powers
# using Sieve Of Eratosthenes

# Using SieveOfEratosthenes to
# find smallest prime factor
# of all the numbers.

# For example, if N is 10,
# s[2] = s[4] = s[6] = s[10] = 2
# s[3] = s[9] = 3
# s[5] = 5
# s[7] = 7
def sieveOfEratosthenes(N, s):

    # Create a boolean array
    # "prime[0..n]" and initialize
    # all entries in it as false.
    prime = [False] * (N+1)

    # Initializing smallest factor
    # equal to 2 for all the even
    # numbers
    for i in range(2, N+1, 2):
        s[i] = 2

    # For odd numbers less than
    # equal to n
    for i in range(3, N+1, 2):
        if (prime[i] == False):

            # s(i) for a prime is
            # the number itself
            s[i] = i

            # For all multiples of
            # current prime number
            for j in range(i, int(N / i) + 1, 2):
                if (prime[i*j] == False):
                    prime[i*j] = True

                    # i is the smallest
                    # prime factor for
                    # number "i*j".
                    s[i * j] = i

# Function to generate prime
# factors and its power
def generatePrimeFactors(N):

    # s[i] is going to store
    # smallest prime factor
    # of i.
    s = [0] * (N+1)

    # Filling values in s[]
    # using sieve
    sieveOfEratosthenes(N, s)

    print("Factor Power")

    # Current prime factor of N
    curr = s[N]

    # Power of current prime factor
    cnt = 1

    # Printing prime factors and
    #their powers
    while (N > 1):
        N //= s[N]

        # N is now N/s[N]. If new N
        # also has smallest prime
        # factor as curr, increment
        # power
        if (curr == s[N]):
            cnt += 1
            continue

        print(str(curr) + "\t" + str(cnt))

        # Update current prime factor
        # as s[N] and initializing
        # count as 1.
        curr = s[N]
        cnt = 1

#Driver Program
N = 360
generatePrimeFactors(N)

# This code is contributed by Ansu Kumari

C#

// C# Program to print prime
// factors and their powers using
// Sieve Of Eratosthenes
class GFG
{
// Using SieveOfEratosthenes
// to find smallest prime
// factor of all the numbers.
// For example, if N is 10,
// s[2] = s[4] = s[6] = s[10] = 2
// s[3] = s[9] = 3
// s[5] = 5
// s[7] = 7
static void sieveOfEratosthenes(int N, int[] s)
{
    // Create a boolean array
    // "prime[0..n]" and initialize
    // all entries in it as false.
    bool[] prime = new bool[N + 1];

    // Initializing smallest
    // factor equal to 2
    // for all the even numbers
    for (int i = 2; i <= N; i += 2)
        s[i] = 2;

    // For odd numbers less
    // then equal to n
    for (int i = 3; i <= N; i += 2)
    {
        if (prime[i] == false)
        {
            // s(i) for a prime is
            // the number itself
            s[i] = i;

            // For all multiples of
            // current prime number
            for (int j = i; j * i <= N; j += 2)
            {
                if (prime[i * j] == false)
                {
                    prime[i * j] = true;

                    // i is the smallest prime
                    // factor for number "i*j".
                    s[i * j] = i;
                }
            }
        }
    }
}

// Function to generate prime
// factors and its power
static void generatePrimeFactors(int N)
{
    // s[i] is going to store
    // smallest prime factor of i.
    int[] s = new int[N + 1];

    // Filling values in s[] using sieve
    sieveOfEratosthenes(N, s);

    System.Console.WriteLine("Factor Power");

    int curr = s[N]; // Current prime factor of N
    int cnt = 1; // Power of current prime factor

    // Printing prime factors
    // and their powers
    while (N > 1)
    {
        N /= s[N];

        // N is now N/s[N]. If new N
        // also has smallest prime
        // factor as curr, increment power
        if (curr == s[N])
        {
            cnt++;
            continue;
        }

        System.Console.WriteLine(curr + "\t" + cnt);

        // Update current prime factor
        // as s[N] and initializing
        // count as 1.
        curr = s[N];
        cnt = 1;
    }
}

// Driver Code
static void Main()
{
    int N = 360;
    generatePrimeFactors(N);
}
}

// This code is contributed by mits

PHP

<?php
// PHP Program to print prime factors and
// their powers using Sieve Of Eratosthenes

// Using SieveOfEratosthenes to find smallest
// prime factor of all the numbers.
// For example, if N is 10,
// s[2] = s[4] = s[6] = s[10] = 2
// s[3] = s[9] = 3
// s[5] = 5
// s[7] = 7
function sieveOfEratosthenes($N, &$s)
{
    // Create a boolean array "prime[0..n]" and
    // initialize all entries in it as false.
    $prime = array_fill(0, $N + 1, false);

    // Initializing smallest factor equal
    // to 2 for all the even numbers
    for ($i = 2; $i <= $N; $i += 2)
        $s[$i] = 2;

    // For odd numbers less than equal to n
    for ($i = 3; $i <= $N; $i += 2)
    {
        if ($prime[$i] == false)
        {
            // s(i) for a prime is the
            // number itself
            $s[$i] = $i;

            // For all multiples of current
            // prime number
            for ($j = $i; $j * $i <= $N; $j += 2)
            {
                if ($prime[$i * $j] == false)
                {
                    $prime[$i * $j] = true;

                    // i is the smallest prime factor
                    // for number "i*j".
                    $s[$i * $j] = $i;
                }
            }
        }
    }
}

// Function to generate prime factors
// and its power
function generatePrimeFactors($N)
{
    // s[i] is going to store smallest
    // prime factor of i.
    $s = array_fill(0, $N + 1, 0);

    // Filling values in s[] using sieve
    sieveOfEratosthenes($N, $s);

    print("Factor Power\n");

    $curr = $s[$N]; // Current prime factor of N
    $cnt = 1; // Power of current prime factor

    // Printing prime factors and their powers
    while ($N > 1)
    {
        if($s[$N])
        $N = (int)($N / $s[$N]);

        // N is now N/s[N]. If new N als has smallest
        // prime factor as curr, increment power
        if ($curr == $s[$N])
        {
            $cnt++;
            continue;
        }

        print($curr . "\t" . $cnt . "\n");

        // Update current prime factor as s[N]
        // and initializing count as 1.
        $curr = $s[$N];
        $cnt = 1;
    }
}

// Driver Code
$N = 360;
generatePrimeFactors($N);

// This code is contributed by mits
?>

Javascript

<script>

// javascript Program to print prime
// factors and their powers using
// Sieve Of Eratosthenes

// Using SieveOfEratosthenes
// to find smallest prime
// factor of all the numbers.
// For example, if N is 10,
// s[2] = s[4] = s[6] = s[10] = 2
// s[3] = s[9] = 3
// s[5] = 5
// s[7] = 7
function sieveOfEratosthenes(N,  s)
{
    // Create a boolean array
    // "prime[0..n]"  and initialize
    // all entries in it as false.
    prime = Array.from({length: N+1}, (_, i) => false);

    // Initializing smallest
    // factor equal to 2
    // for all the even numbers
    for (i = 2; i <= N; i += 2)
        s[i] = 2;

    // For odd numbers less
    // then equal to n
    for (i = 3; i <= N; i += 2)
    {
        if (prime[i] == false)
        {
            // s(i) for a prime is
            // the number itself
            s[i] = i;

            // For all multiples of
            // current prime number
            for (j = i; j * i <= N; j += 2)
            {
                if (prime[i * j] == false)
                {
                    prime[i * j] = true;

                    // i is the smallest prime
                    // factor for number "i*j".
                    s[i * j] = i;
                }
            }
        }
    }
}

// Function to generate prime
// factors and its power
function generatePrimeFactors(N)
{
    // s[i] is going to store
    // smallest prime factor of i.
    var s = Array.from({length: N+1}, (_, i) => 0);

    // Filling values in s using sieve
    sieveOfEratosthenes(N, s);

    document.write("Factor Power");

    var curr = s[N]; // Current prime factor of N
    var cnt = 1; // Power of current prime factor

    // Printing prime factors
    // and their powers
    while (N > 1)
    {
        N /= s[N];

        // N is now N/s[N]. If new N
        // also has smallest prime
        // factor as curr, increment power
        if (curr == s[N])
        {
            cnt++;
            continue;
        }

        document.write("<br>"+curr + "\t" + cnt);

        // Update current prime factor
        // as s[N] and initializing
        // count as 1.
        curr = s[N];
        cnt = 1;
    }
}

// Driver Code
var N = 360;
generatePrimeFactors(N);


// This code contributed by Princi Singh
</script>

Output:

Factor  Power
  2      3
  3      2
  5      1

Time Complexity: O(n*log(log(n)))
Auxiliary Space: O(n)

The above algorithm finds all powers in O(Log N) time after we have filled s[]. This can be very useful in competitive environment where we have an upper limit and we need to compute prime factors and their powers for many test cases. In this scenario, the array needs to be s[] filled only once.
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