Ordered Prime Signature

Given a number n, find the ordered prime signatures and using this find the number of divisor of given n. 
Any positive integer, ‘n’ can be expressed in the form of its prime factors. If ‘n’ has p₁, p₂, … etc. as its prime factors, then n can be expressed as : 

Now, arrange the obtained exponents of the prime factors of ‘n’ in non-decreasing order. The arrangement thus obtained is called the ordered prime signature of the positive integer ‘n’.
Example:

Input : n = 20
Output :  
The Ordered Prime Signature of 20 is : 
{ 1, 2 }
The total number of divisors of 20 is 6

Input : n = 13
Output :  
The Ordered Prime Signature of 13 is : 
{ 1 }
The total number of divisors of 13 is 2

Explanation :

1. ordered prime signature of 20 = { 1, 2 }
2. ordered prime signature of 37 = { 1 }
3. ordered prime signature of 49 = { 2 }

It can be ascertained from the above discussion that the prime signature of 1 is { 1 }. Furthermore, all prime numbers have the same signature, i.e { 1 } and the prime signature of a number, that is the k-th power of a prime number (say, 25 which is the 2-nd power of 5), is always { k }.
For example : 

Ordered Prime signature of 100 = { 2, 2 }, as 100 = 2^2 × 5^2 
Now adding one to each element gives { 3, 3 } and the product is 3 × 3 = 9, 
i.e the total number of divisors of 100 is nine. 
They are 1, 2, 4, 5, 10, 20, 25, 50, 100. 

Approach :
1) Find the prime factorization of the number 
2) Store each exponent corresponding to a prime factor, in a vector 
3) Sort the vector in ascending order 
4) Add one to each element present in the vector 
5) Multiply all the elements 

C++

// CPP to find total number of divisors of a
// number, using ordered prime signature
#include <bits/stdc++.h>
using namespace std;

// Finding primes upto entered number
vector<int> primes(int n)
{
    bool prime[n + 1];

    // Finding primes by Sieve
    // of Eratosthenes method
    memset(prime, true, sizeof(prime));

    for (int i = 2; i * i <= n; i++)
    {

        // If prime[i] is not changed,
        // then it is prime
        if (prime[i] == true) {

            // Update all multiples of p
            for (int j = i * 2; j <= n; j += i)
                prime[j] = false;
        }
    }

    vector<int> arr;

    // Forming array of the prime numbers found
    for (int i = 2; i <= n; i++)
    {
        if (prime[i])
            arr.push_back(i);
    }
    return arr;
}

// Finding ordered prime signature of the number
vector<int> signature( int n)
{
    vector<int> r = primes(n);

    // Map to store prime factors and
    // the related exponents
    map<int, int> factor;

    // Declaring an iterator for map
    map<int, int>::iterator it;
    vector<int> sort_exp;
    int k, t = n;
    it = factor.begin();

    // Finding prime factorization of the number
    for (int i = 0; i < r.size(); i++)
    {
        if (n % r[i] == 0) {
            k = 0;
            while (n % r[i] == 0) {
                n = n / r[i];
                k++;
            }

            // Storing the prime factor and
            // its exponent in map
            factor.insert(it, pair<int, int>(r[i], k));

            // Storing the exponent in a vector
            sort_exp.push_back(k);
        }
    }

    // Sorting the stored exponents
    sort(sort_exp.begin(), sort_exp.end());

    // Printing the prime signature
    cout << " The Ordered Prime Signature of " <<
         t << " is : \n{ ";

    for (int i = 0; i < sort_exp.size(); i++)
    {
        if (i != sort_exp.size() - 1)
            cout << sort_exp[i] << ", ";
        else
            cout << sort_exp[i] << " }\n";
    }
    return sort_exp;
}

// Finding total number of divisors of the number
void divisors(int n)
{
    int f = 1, l;
    vector<int> div = signature(n);
    l = div.size();

    // Adding one to each element present
    for (int i = 0; i < l; i++)
    {

        // in ordered prime signature
        div[i] += 1;

        // Multiplying the elements
        f *= div[i];
    }
    cout << "The total number of divisors of " <<
          n << " is " << f << "\n";
}

// Driver Method
int main()
{
    int n = 13;
    divisors(n);
    return 0;
}

Java

// JAVA to find total number of divisors of a
// number, using ordered prime signature
import java.util.*;

class GFG
{
    static class pair
    {
        int first, second;
        public pair(int first, int second)
        {
            this.first = first;
            this.second = second;
        }
    }

// Finding primes upto entered number
static Vector<Integer> primes(int n)
{
    boolean []prime = new boolean[n + 1];

    // Finding primes by Sieve
    // of Eratosthenes method
    Arrays.fill(prime, true);

    for (int i = 2; i * i <= n; i++)
    {

        // If prime[i] is not changed,
        // then it is prime
        if (prime[i] == true) {

            // Update all multiples of p
            for (int j = i * 2; j <= n; j += i)
                prime[j] = false;
        }
    }
    Vector<Integer> arr = new Vector<>();

    // Forming array of the prime numbers found
    for (int i = 2; i <= n; i++)
    {
        if (prime[i])
            arr.add(i);
    }
    return arr;
}

// Finding ordered prime signature of the number
static Vector<Integer> signature( int n)
{
    Vector<Integer> r = primes(n);

    // Map to store prime factors and
    // the related exponents
    HashMap<Integer,Integer> factor = new HashMap<>();

    // Declaring an iterator for map
  //  HashMap<Integer,Integer>::iterator it;
    Vector<Integer> sort_exp = new Vector<>();
    int k, t = n;
    int it = 0;

    // Finding prime factorization of the number
    for (int i = 0; i < r.size(); i++)
    {
        if (n % r.get(i) == 0)
        {
            k = 0;
            while (n % r.get(i) == 0)
            {
                n = n / r.get(i);
                k++;
            }

            // Storing the prime factor and
            // its exponent in map
            factor.put(r.get(i), k);

            // Storing the exponent in a vector
            sort_exp.add(k);
        }
    }

    // Sorting the stored exponents
    Collections.sort(sort_exp);

    // Printing the prime signature
    System.out.print(" The Ordered Prime Signature of " +
         t+ " is : \n{ ");

    for (int i = 0; i < sort_exp.size(); i++)
    {
        if (i != sort_exp.size() - 1)
            System.out.print(sort_exp.get(i) + ", ");
        else
            System.out.print(sort_exp.get(i) + " }\n");
    }
    return sort_exp;
}

// Finding total number of divisors of the number
static void divisors(int n)
{
    int f = 1, l;
    Vector<Integer> div = signature(n);
    l = div.size();

    // Adding one to each element present
    for (int i = 0; i < l; i++)
    {

        // in ordered prime signature
        //div[i] += 1;

        // Multiplying the elements
        f *= (div.get(i) + 1);
    }
    System.out.print("The total number of divisors of " +
          n + " is " +  f + "\n");
}

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

// This code is contributed by aashish1995

Python3

# Python3 to find total number of divisors of a
# number, using ordered prime signature

# Finding primes upto entered number
def primes(n):

    # Finding primes by Sieve
    # of Eratosthenes method
    prime = [True for _ in range(n + 1)]

    for i in range(2, int(n ** 0.5) + 1):

        # If prime[i] is not changed,
        # then it is prime
        if (prime[i] == True):

            # Update all multiples of p
            for j in range(2 * i, n + 1, i):
                prime[j] = False

    arr = []

    # Forming array of the prime numbers found
    for i in range(2, n + 1):

        if (prime[i]):
            arr.append(i)

    return arr


# Finding ordered prime signature of the number
def signature(n):
    r = primes(n)

    # Map to store prime factors and
    # the related exponents
    factor = dict()

    sort_exp = []
    t = n

    # Finding prime factorization of the number
    for i in range(len(r)):
        if (n % r[i] == 0):
            k = 0
            while (n % r[i] == 0):
                n //= r[i]
                k = k + 1

            # Storing the prime factor and
            # its exponent in map
            factor[r[i]] = k

            # Storing the exponent in a vector
            sort_exp.append(k)

    # Sorting the stored exponents
    sort_exp.sort()

    # Printing the prime signature
    print(" The Ordered Prime Signature of ", t, " is :")
    print("{", end=" ")
    for i in range(len(sort_exp)):

        if (i != len(sort_exp) - 1):
            print(sort_exp[i], end=", ")
        else:
            print(sort_exp[i], "}")

    return sort_exp


# Finding total number of divisors of the number
def divisors(n):
    f = 1
    div = signature(n)
    l = len(div)

    # Adding one to each element present
    for i in range(l):
        # in ordered prime signature
        div[i] += 1

        # Multiplying the elements
        f *= div[i]

    print("The total number of divisors of ", n, " is ", f)


# Driver Method
n = 13
divisors(n)

# This code is contributed by phasing17

C#

// C# to find total number
// of divisors of a number,
// using ordered prime signature
using System;
using System.Collections.Generic;

class GFG
{
    // Finding primes
    // upto entered number
    static List<int> primes(int n)
    {
        bool []prime = new bool[n + 1];

        // Finding primes by Sieve
        // of Eratosthenes method
        for (int i = 0; i < n + 1; i++)
            prime[i] = true;

        for (int i = 2; i * i <= n; i++)
        {

            // If prime[i] is not
            // changed, then it is prime
            if (prime[i] == true)
            {

                // Update all multiples of p
                for (int j = i * 2;
                         j <= n; j += i)
                    prime[j] = false;
            }
        }

        List<int> arr = new List<int>();

        // Forming array of the
        // prime numbers found
        for (int i = 2; i <= n; i++)
        {
            if (prime[i])
                arr.Add(i);
        }
        return arr;
    }

    // Finding ordered prime
    // signature of the number
    static List<int> signature( int n)
    {
        List<int> r = primes(n);

        // Map to store prime factors
        // and the related exponents
        var factor = new Dictionary<int, int>();

        List<int> sort_exp = new List<int>();
        int k, t = n;

        // Finding prime factorization
        // of the number
        for (int i = 0; i < r.Count; i++)
        {
            if (n % r[i] == 0)
            {
                k = 0;
                while (n % r[i] == 0)
                {
                    n = n / r[i];
                    k++;
                }

                // Storing the prime factor
                // and its exponent in map
                factor.Add(r[i], k);

                // Storing the exponent
                // in a List
                sort_exp.Add(k);
            }
        }

        // Sorting the
        // stored exponents
        sort_exp.Sort();

        // Printing the
        // prime signature
        Console.Write(" The Ordered Prime Signature of " +
                                        t + " is : \n{ ");

        for (int i = 0; i < sort_exp.Count; i++)
        {
            if (i != sort_exp.Count - 1)
                Console.Write(sort_exp[i] + ", ");
            else
                Console.Write(sort_exp[i] + " }\n");
        }
        return sort_exp;
    }

    // Finding total number
    // of divisors of the number
    static void divisors(int n)
    {
        int f = 1, l;
        List<int> div = signature(n);
        l = div.Count;

        // Adding one to each
        // element present
        for (int i = 0; i < l; i++)
        {

            // in ordered
            // prime signature
            div[i] += 1;

            // Multiplying
            // the elements
            f *= div[i];
        }
        Console.Write("The total number of divisors of " +
                                   n + " is " + f + "\n");
    }

    // Driver Code
    static void Main()
    {
        int n = 13;
        divisors(n);
    }
}

// This code is contributed by
// Manish Shaw(manishshaw1)

Javascript

<script>
// JavaScript to find total number of divisors of a
// number, using ordered prime signature

// Finding primes upto entered number
function primes(n)
{
    // Finding primes by Sieve
    // of Eratosthenes method
    let prime = new Array(n+1).fill(true);

    for (let i = 2; i * i <= n; i++)
    {

        // If prime[i] is not changed,
        // then it is prime
        if (prime[i] == true) {

            // Update all multiples of p
            for (let j = i * 2; j <= n; j += i)
                prime[j] = false;
        }
    }

    let arr = new Array();

    // Forming array of the prime numbers found
    for (let i = 2; i <= n; i++)
    {
        if (prime[i])
            arr.push(i);
    }
    return arr;
}

// Finding ordered prime signature of the number
function signature(n)
{
    let r = primes(n);

    // Map to store prime factors and
    // the related exponents
    let factor = new Map();

    let sort_exp = new Array();
    let k;
    let t = n;

    // Finding prime factorization of the number
    for (let i = 0; i < r.length; i++)
    {
        if (n % r[i] == 0) {
            k = 0;
            while (n % r[i] == 0) {
                n = Math.floor(n / r[i]);
                k = k + 1;
            }

            // Storing the prime factor and
            // its exponent in map
            factor.set(r[i], k);

            // Storing the exponent in a vector
            sort_exp.push(k);
        }
    }

    // Sorting the stored exponents
    sort_exp.sort();

    // Printing the prime signature
    document.write(" The Ordered Prime Signature of ", t, " is :\n");
    document.write("{");
    for (let i = 0; i < sort_exp.length; i++)
    {
        if (i != sort_exp.length - 1)
            document.write(sort_exp[i],",");
        else
            document.write(sort_exp[i], " } \n");
    }
    return sort_exp;
}

// Finding total number of divisors of the number
function divisors(n)
{
    let f = 1;
    let l;
    let div = signature(n);
    l = div.length;

    // Adding one to each element present
    for (let i = 0; i < l; i++)
    {

        // in ordered prime signature
        div[i] += 1;

        // Multiplying the elements
        f *= div[i];
    }
    document.write("The total number of divisors of ", n, " is ", f);
}

// Driver Method
let n = 13;
divisors(n);

// This code is contributed by gautamgoel962.
</script>

Output:

The Ordered Prime Signature of 13 is : 
{ 1 }
The total number of divisors of 13 is 2

Application :
Finding the ordered prime signature of a number has utilities in finding number of divisors. In fact, the total number of divisors of a number can be inferred from the ordered prime signature of that number. To accomplish this, just add one to each element present in the ordered prime signature and then multiply those elements. The product, thus obtained gives the total number of divisors of the number (including 1 and the number itself).