Algorithm to calculate the number of divisors of a given number

StackOverflow https://stackoverflow.com/questions/110344

Question

What would be the most optimal algorithm (performance-wise) to calculate the number of divisors of a given number?

It'll be great if you could provide pseudocode or a link to some example.

EDIT: All the answers have been very helpful, thank you. I'm implementing the Sieve of Atkin and then I'm going to use something similar to what Jonathan Leffler indicated. The link posted by Justin Bozonier has further information on what I wanted.

Was it helpful?

Solution

Dmitriy is right that you'll want the Sieve of Atkin to generate the prime list but I don't believe that takes care of the whole issue. Now that you have a list of primes you'll need to see how many of those primes act as a divisor (and how often).

Here's some python for the algo Look here and search for "Subject: math - need divisors algorithm". Just count the number of items in the list instead of returning them however.

Here's a Dr. Math that explains what exactly it is you need to do mathematically.

Essentially it boils down to if your number n is:
n = a^x * b^y * c^z
(where a, b, and c are n's prime divisors and x, y, and z are the number of times that divisor is repeated) then the total count for all of the divisors is:
(x + 1) * (y + 1) * (z + 1).

Edit: BTW, to find a,b,c,etc you'll want to do what amounts to a greedy algo if I'm understanding this correctly. Start with your largest prime divisor and multiply it by itself until a further multiplication would exceed the number n. Then move to the next lowest factor and times the previous prime ^ number of times it was multiplied by the current prime and keep multiplying by the prime until the next will exceed n... etc. Keep track of the number of times you multiply the divisors together and apply those numbers into the formula above.

Not 100% sure about my algo description but if that isn't it it's something similar .

OTHER TIPS

There are a lot more techniques to factoring than the sieve of Atkin. For example suppose we want to factor 5893. Well its sqrt is 76.76... Now we'll try to write 5893 as a product of squares. Well (77*77 - 5893) = 36 which is 6 squared, so 5893 = 77*77 - 6*6 = (77 + 6)(77-6) = 83*71. If that hadn't worked we'd have looked at whether 78*78 - 5893 was a perfect square. And so on. With this technique you can quickly test for factors near the square root of n much faster than by testing individual primes. If you combine this technique for ruling out large primes with a sieve, you will have a much better factoring method than with the sieve alone.

And this is just one of a large number of techniques that have been developed. This is a fairly simple one. It would take you a long time to learn, say, enough number theory to understand the factoring techniques based on elliptic curves. (I know they exist. I don't understand them.)

Therefore unless you are dealing with small integers, I wouldn't try to solve that problem myself. Instead I'd try to find a way to use something like the PARI library that already has a highly efficient solution implemented. With that I can factor a random 40 digit number like 124321342332143213122323434312213424231341 in about .05 seconds. (Its factorization, in case you wondered, is 29*439*1321*157907*284749*33843676813*4857795469949. I am quite confident that it didn't figure this out using the sieve of Atkin...)

@Yasky

Your divisors function has a bug in that it does not work correctly for perfect squares.

Try:

int divisors(int x) {
    int limit = x;
    int numberOfDivisors = 0;

    if (x == 1) return 1;

    for (int i = 1; i < limit; ++i) {
        if (x % i == 0) {
            limit = x / i;
            if (limit != i) {
                numberOfDivisors++;
            }
            numberOfDivisors++;
        }
    }

    return numberOfDivisors;
}

I disagree that the sieve of Atkin is the way to go, because it could easily take longer to check every number in [1,n] for primality than it would to reduce the number by divisions.

Here's some code that, although slightly hackier, is generally much faster:

import operator
# A slightly efficient superset of primes.
def PrimesPlus():
  yield 2
  yield 3
  i = 5
  while True:
    yield i
    if i % 6 == 1:
      i += 2
    i += 2
# Returns a dict d with n = product p ^ d[p]
def GetPrimeDecomp(n):
  d = {}
  primes = PrimesPlus()
  for p in primes:
    while n % p == 0:
      n /= p
      d[p] = d.setdefault(p, 0) + 1
    if n == 1:
      return d
def NumberOfDivisors(n):
  d = GetPrimeDecomp(n)
  powers_plus = map(lambda x: x+1, d.values())
  return reduce(operator.mul, powers_plus, 1)

ps That's working python code to solve this problem.

This interesting question is much harder than it looks, and it has not been answered. The question can be factored into 2 very different questions.

1 given N, find the list L of N's prime factors

2 given L, calculate number of unique combinations

All answers I see so far refer to #1 and fail to mention it is not tractable for enormous numbers. For moderately sized N, even 64-bit numbers, it is easy; for enormous N, the factoring problem can take "forever". Public key encryption depends on this.

Question #2 needs more discussion. If L contains only unique numbers, it is a simple calculation using the combination formula for choosing k objects from n items. Actually, you need to sum the results from applying the formula while varying k from 1 to sizeof(L). However, L will usually contain multiple occurrences of multiple primes. For example, L = {2,2,2,3,3,5} is the factorization of N = 360. Now this problem is quite difficult!

Restating #2, given collection C containing k items, such that item a has a' duplicates, and item b has b' duplicates, etc. how many unique combinations of 1 to k-1 items are there? For example, {2}, {2,2}, {2,2,2}, {2,3}, {2,2,3,3} must each occur once and only once if L = {2,2,2,3,3,5}. Each such unique sub-collection is a unique divisor of N by multiplying the items in the sub-collection.

An answer to your question depends greatly on the size of the integer. Methods for small numbers, e.g. less then 100 bit, and for numbers ~1000 bit (such as used in cryptography) are completely different.

Here is a straight forward O(sqrt(n)) algorithm. I used this to solve project euler

def divisors(n):
    count=2 # accounts for 'n' and '1'
    i=2
    while(i**2 < n):
        if(n%i==0):
            count+=2
        i+=1
    count+=(1 if i**2==n else 0)
    return count

JUST one line
I have thought very carefuly about your question and I have tried to write a highly efficient and performant piece of code To print all divisors of a given number on screen we need just one line of code! (use option -std=c99 while compiling via gcc)

for(int i=1,n=9;((!(n%i)) && printf("%d is a divisor of %d\n",i,n)) || i<=(n/2);i++);//n is your number

for finding numbers of divisors you can use the following very very fast function(work correctly for all integer number except 1 and 2)

int number_of_divisors(int n)
{
    int counter,i;
    for(counter=0,i=1;(!(n%i) && (counter++)) || i<=(n/2);i++);
    return counter;
}

or if you treat given number as a divisor(work correctly for all integer number except 1 and 2) 

int number_of_divisors(int n)
{
    int counter,i;
    for(counter=0,i=1;(!(n%i) && (counter++)) || i<=(n/2);i++);
    return ++counter;
}

NOTE:two above functions works correctly for all positive integer number except number 1 and 2 so it is functional for all numbers that are greater than 2 but if you Need to cover 1 and 2 , you can use one of the following functions( a little slower)

int number_of_divisors(int n)
{
    int counter,i;
    for(counter=0,i=1;(!(n%i) && (counter++)) || i<=(n/2);i++);
    if (n==2 || n==1)
    {
    return counter;
    }
    return ++counter;
}

OR

int number_of_divisors(int n)
{
    int counter,i;
for(counter=0,i=1;(!(i==n) && !(n%i) && (counter++)) || i<=(n/2);i++);
    return ++counter;
}

small is beautiful :)

The sieve of Atkin is an optimized version of the sieve of Eratosthenes which gives all prime numbers up to a given integer. You should be able to google this for more detail.

Once you have that list, it's a simple matter to divide your number by each prime to see if it's an exact divisor (i.e., remainder is zero).

The basic steps calculating the divisors for a number (n) are [this is pseudocode converted from real code so I hope I haven't introduced errors]:

for z in 1..n:
    prime[z] = false
prime[2] = true;
prime[3] = true;

for x in 1..sqrt(n):
    xx = x * x

    for y in 1..sqrt(n):
        yy = y * y

        z = 4*xx+yy
        if (z <= n) and ((z mod 12 == 1) or (z mod 12 == 5)):
            prime[z] = not prime[z]

        z = z-xx
        if (z <= n) and (z mod 12 == 7):
            prime[z] = not prime[z]

        z = z-yy-yy
        if (z <= n) and (x > y) and (z mod 12 == 11):
            prime[z] = not prime[z]

for z in 5..sqrt(n):
    if prime[z]:
        zz = z*z
        x = zz
        while x <= limit:
            prime[x] = false
            x = x + zz

for z in 2,3,5..n:
    if prime[z]:
        if n modulo z == 0 then print z

You might try this one. It's a bit hackish, but it's reasonably fast.

def factors(n):
    for x in xrange(2,n):
        if n%x == 0:
            return (x,) + factors(n/x)
    return (n,1)

Once you have the prime factorization, there is a way to find the number of divisors. Add one to each of the exponents on each individual factor and then multiply the exponents together.

For example: 36 Prime Factorization: 2^2*3^2 Divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36 Number of Divisors: 9

Add one to each exponent 2^3*3^3 Multiply exponents: 3*3 = 9

Before you commit to a solution consider that the Sieve approach might not be a good answer in the typical case.

A while back there was a prime question and I did a time test--for 32-bit integers at least determining if it was prime was slower than brute force. There are two factors going on:

1) While a human takes a while to do a division they are very quick on the computer--similar to the cost of looking up the answer.

2) If you do not have a prime table you can make a loop that runs entirely in the L1 cache. This makes it faster.

This is an efficient solution:

#include <iostream>
int main() {
  int num = 20; 
  int numberOfDivisors = 1;

  for (int i = 2; i <= num; i++)
  {
    int exponent = 0;
    while (num % i == 0) {
        exponent++; 
        num /= i;
    }   
    numberOfDivisors *= (exponent+1);
  }

  std::cout << numberOfDivisors << std::endl;
  return 0;
}

Divisors do something spectacular: they divide completely. If you want to check the number of divisors for a number, n, it clearly is redundant to span the whole spectrum, 1...n. I have not done any in-depth research for this but I solved Project Euler's problem 12 on Triangular Numbers. My solution for the greater then 500 divisors test ran for 309504 microseconds (~0.3s). I wrote this divisor function for the solution.

int divisors (int x) {
    int limit = x;
    int numberOfDivisors = 1;

    for (int i(0); i < limit; ++i) {
        if (x % i == 0) {
            limit = x / i;
            numberOfDivisors++;
        }
    }

    return numberOfDivisors * 2;
}

To every algorithm, there is a weak point. I thought this was weak against prime numbers. But since triangular numbers are not print, it served its purpose flawlessly. From my profiling, I think it did pretty well.

Happy Holidays.

You want the Sieve of Atkin, described here: http://en.wikipedia.org/wiki/Sieve_of_Atkin

the prime number method is very clear here . P[] is a list of prime number less than or equal the sq = sqrt(n) ;

for (int i = 0 ; i < size && P[i]<=sq ; i++){
          nd = 1;
          while(n%P[i]==0){
               n/=P[i];
               nd++;
               }
          count*=nd;
          if (n==1)break;
          }
      if (n!=1)count*=2;//the confusing line :D :P .

     i will lift the understanding for the reader  .
     i now look forward to a method more optimized  .

Number theory textbooks call the divisor-counting function tau. The first interesting fact is that it's multiplicative, ie. τ(ab) = τ(a)τ(b) , when a and b have no common factor. (Proof: each pair of divisors of a and b gives a distinct divisor of ab).

Now note that for p a prime, τ(p**k) = k+1 (the powers of p). Thus you can easily compute τ(n) from its factorisation.

However factorising large numbers can be slow (the security of RSA crytopraphy depends on the product of two large primes being hard to factorise). That suggests this optimised algorithm

  1. Test if the number is prime (fast)
  2. If so, return 2
  3. Otherwise, factorise the number (slow if multiple large prime factors)
  4. Compute τ(n) from the factorisation

The following is a C program to find the number of divisors of a given number.

The complexity of the above algorithm is O(sqrt(n)).

This algorithm will work correctly for the number which are perfect square as well as the numbers which are not perfect square.

Note that the upperlimit of the loop is set to the square-root of number to have the algorithm most efficient.

Note that storing the upperlimit in a separate variable also saves the time, you should not call the sqrt function in the condition section of the for loop, this also saves your computational time.

#include<stdio.h>
#include<math.h>
int main()
{
    int i,n,limit,numberOfDivisors=1;
    printf("Enter the number : ");
    scanf("%d",&n);
    limit=(int)sqrt((double)n);
    for(i=2;i<=limit;i++)
        if(n%i==0)
        {
            if(i!=n/i)
                numberOfDivisors+=2;
            else
                numberOfDivisors++;
        }
    printf("%d\n",numberOfDivisors);
    return 0;
}

Instead of the above for loop you can also use the following loop which is even more efficient as this removes the need to find the square-root of the number.

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

Here is a function that I wrote. it's worst time complexity is O(sqrt(n)),best time on the other hand is O(log(n)). It gives you all the prime divisors along with the number of its occurence.

public static List<Integer> divisors(n) {   
    ArrayList<Integer> aList = new ArrayList();
    int top_count = (int) Math.round(Math.sqrt(n));
    int new_n = n;

    for (int i = 2; i <= top_count; i++) {
        if (new_n == (new_n / i) * i) {
            aList.add(i);
            new_n = new_n / i;
            top_count = (int) Math.round(Math.sqrt(new_n));
            i = 1;
        }
    }
    aList.add(new_n);
    return aList;
}

This is the most basic way of computing the number divissors:

class PrintDivisors
{
    public static void main(String args[])
    {

    System.out.println("Enter the number");

    // Create Scanner object for taking input
    Scanner s=new Scanner(System.in);

    // Read an int
    int n=s.nextInt();

        // Loop from 1 to 'n'
        for(int i=1;i<=n;i++)
        {

            // If remainder is 0 when 'n' is divided by 'i',
            if(n%i==0)
            {
            System.out.print(i+", ");
            }
        }

    // Print [not necessary]    
    System.out.print("are divisors of "+n);

    }
}

@Kendall

I tested your code and made some improvements, now it is even faster. I also tested with @هومن جاویدپور code, this is also faster than his code.

long long int FindDivisors(long long int n) {
  long long int count = 0;
  long long int i, m = (long long int)sqrt(n);
  for(i = 1;i <= m;i++) {
    if(n % i == 0)
      count += 2;
  }
  if(n / m == m && n % m == 0)
    count--;
  return count;
}

Isn't this just a question of factoring the number - determining all the factors of the number? You can then decide whether you need all combinations of one or more factors.

So, one possible algorithm would be:

factor(N)
    divisor = first_prime
    list_of_factors = { 1 }
    while (N > 1)
        while (N % divisor == 0)
            add divisor to list_of_factors
            N /= divisor
        divisor = next_prime
    return list_of_factors

It is then up to you to combine the factors to determine the rest of the answer.

This is something i came up with based on Justin answer. It might require some optimization.

n=int(input())

a=[]
b=[]

def sieve(n):
    np = n + 1
    s = list(range(np)) 
    s[1] = 0
    sqrtn = int(n**0.5)
    for i in range(2, sqrtn + 1): 
        if s[i]:
            s[i*i: np: i] = [0] * len(range(i*i, np, i))
    return filter(None, s)

k=list(sieve(n))

for i in range(len(k)):
        if n%k[i]==0:
                a.append(k[i])

a.sort()

for i in range(len(a)):
        j=1
        while n%(a[i]**j)==0: 
                j=j+1
        b.append(j-1)

nod=1

for i in range(len(b)):
        nod=nod*(b[i]+1)

print('no.of divisors of {} = {}'.format(n,nod))

I think this is what you are looking for.I does exactly what you asked for. Copy and Paste it in Notepad.Save as *.bat.Run.Enter Number.Multiply the process by 2 and thats the number of divisors.I made that on purpose so the it determine the divisors faster:

Pls note that a CMD varriable cant support values over 999999999

@echo off

modecon:cols=100 lines=100

:start
title Enter the Number to Determine 
cls
echo Determine a number as a product of 2 numbers
echo.
echo Ex1 : C = A * B
echo Ex2 : 8 = 4 * 2
echo.
echo Max Number length is 9
echo.
echo If there is only 1 proces done  it
echo means the number is a prime number
echo.
echo Prime numbers take time to determine
echo Number not prime are determined fast
echo.

set /p number=Enter Number : 
if %number% GTR 999999999 goto start

echo.
set proces=0
set mindet=0
set procent=0
set B=%Number%

:Determining

set /a mindet=%mindet%+1

if %mindet% GTR %B% goto Results

set /a solution=%number% %%% %mindet%

if %solution% NEQ 0 goto Determining
if %solution% EQU 0 set /a proces=%proces%+1

set /a B=%number% / %mindet%

set /a procent=%mindet%*100/%B%

if %procent% EQU 100 set procent=%procent:~0,3%
if %procent% LSS 100 set procent=%procent:~0,2%
if %procent% LSS 10 set procent=%procent:~0,1%

title Progress : %procent% %%%



if %solution% EQU 0 echo %proces%. %mindet% * %B% = %number%
goto Determining

:Results

title %proces% Results Found
echo.
@pause
goto start

i guess this one will be handy as well as precise

script.pyton

>>>factors=[ x for x in range (1,n+1) if n%x==0] print len(factors)

Try something along these lines:

int divisors(int myNum) {
    int limit = myNum;
    int divisorCount = 0;
    if (x == 1) 
        return 1;
    for (int i = 1; i < limit; ++i) {
        if (myNum % i == 0) {
            limit = myNum / i;
            if (limit != i)
                divisorCount++;
            divisorCount++;
        }
    }
    return divisorCount;
}

You can precompute primes up to the sqaure root of the max possible N and compute the exponent of every prime factor of a number. The number of divisors of n (n = p1^a p2^b p3^c...) is (a+1)(b+1)(c+1) because it's the same as count the way of combine the prime numbers of this factors (and this will count the number of divisors). This is very fast if you precompute the prime numbers

More detailed info about this method:

https://mathschallenge.net/library/number/number_of_divisors

https://www.math.upenn.edu/~deturck/m170/wk2/numdivisors.html

http://primes.utm.edu/glossary/xpage/tau.html

#include <iostream>
#include <cmath>
#include <algorithm>
#include <vector>
using namespace std;

int divisors_count(const vector<int>& primes, int n)
{
    int divisors = 1;
    for (int i = 0; i < primes.size(); ++i) {
        int factor = primes[i];
        int factor_exponent = 0;
        while (n % factor == 0) {
            ++factor_exponent;
            n /= factor;
        }
        divisors *= (factor_exponent + 1);
    }
    if (n > 1) 
        return 2*divisors; // prime factor > sqrt(MAX_N)
    return divisors;
}

int main()
{
    const int MAX_N = 1e6;
    int max_factor = sqrt(MAX_N);

    vector<char> prime(max_factor + 1, true);
    for (int i = 3; i <= max_factor; i += 2) {
        if (prime[i]) {
            for (int j = 3*i; j <= max_factor; j += 2*i) {
                prime[j] = false;
            }   
        }
    }

    vector<int> primes;
    primes.reserve(max_factor/2);
    primes.push_back(2);
    for (int i = 3; i <= max_factor; i += 2) {
        if (prime[i]) {
            primes.push_back(i);
        }
    }

    int n;
    while (cin >> n) {
        cout << divisors_count(primes, n) << endl;
    }
}

I don't know the MOST efficient method, but I'd do the following:

  • Create a table of primes to find all primes less than or equal to the square root of the number (Personally, I'd use the Sieve of Atkin)
  • Count all primes less than or equal to the square root of the number and multiply that by two. If the square root of the number is an integer, then subtract one from the count variable.

Should work \o/

If you need, I can code something up tomorrow in C to demonstrate.

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