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 :
n = {p_1}^{e1} * {p_2}^{e2} * ...
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. 20 = 2^2 * 5^1, ordered prime signature of 20 = { 1, 2 }
  2. 37 = 37^1, ordered prime signature of 37 = { 1 }
  3. 49 = 7^2, 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

[sourcecode language=”CPP”]
// 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 Seive
// 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;
}
[/sourcecode]

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).



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