Given a number n, print all primes smaller than or equal to n. It is also given that n is a small number.

For example, if n is 10, the output should be “2, 3, 5, 7”. If n is 20, the output should be “2, 3, 5, 7, 11, 13, 17, 19”.

The sieve of Eratosthenes is one of the most efficient ways to find all primes smaller than n when n is smaller than 10 million or so (Ref Wiki).

Following is the algorithm to find all the prime numbers less than or equal to a given integer *n* by Eratosthenes’ method:

- Create a list of consecutive integers from 2 to
*n*: (2, 3, 4, …,*n*). - Initially, let
*p*equal 2, the first prime number. - Starting from
*p*, count up in increments of*p*and mark each of these numbers greater than*p*itself in the list. These numbers will be 2*p*, 3*p*, 4*p*, etc.; note that some of them may have already been marked. - Find the first number greater than
*p*in the list that is not marked. If there was no such number, stop. Otherwise, let*p*now equal this number (which is the next prime), and repeat from step 3.

When the algorithm terminates, all the numbers in the list that are not marked are prime.

**Explanation with Example:**

Let us take an example when n = 50. So we need to print all print numbers smaller than or equal to 50.

We create a list of all numbers from 2 to 50.

According to the algorithm we will mark all the numbers which are divisible by 2.

Now we move to our next unmarked number 3 and mark all the numbers which are multiples of 3.

We move to our next unmarked number 5 and mark all multiples of 5.

We continue this process and our final table will look like below:

So the prime numbers are the unmarked ones: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

Thanks to** Krishan Kumar** for providing above explanation.

Implementation:

Following is C++ implementation of the above algorithm. In the following implementation, a boolean array arr[] of size n is used to mark multiples of prime numbers.

## C/C++

// C++ program to print all primes smaller than or equal to // n using Sieve of Eratosthenes #include <bits/stdc++.h> using namespace std; void SieveOfEratosthenes(int n) { // Create a boolean array "prime[0..n]" and initialize // all entries it as true. A value in prime[i] will // finally be false if i is Not a prime, else true. bool prime[n+1]; memset(prime, true, sizeof(prime)); for (int p=2; p*p<=n; p++) { // If prime[p] is not changed, then it is a prime if (prime[p] == true) { // Update all multiples of p for (int i=p*2; i<=n; i += p) prime[i] = false; } } // Print all prime numbers for (int p=2; p<=n; p++) if (prime[p]) cout << p << " "; } // Driver Program to test above function int main() { int n = 30; cout << "Following are the prime numbers smaller " << " than or equal to " << n << endl; SieveOfEratosthenes(n); return 0; }

## Java

// Java program to print all primes smaller than or equal to // n using Sieve of Eratosthenes class SieveOfEratosthenes { void sieveOfEratosthenes(int n) { // Create a boolean array "prime[0..n]" and initialize // all entries it as true. A value in prime[i] will // finally be false if i is Not a prime, else true. boolean prime[] = new boolean[n+1]; for(int i=0;i<n;i++) prime[i] = true; for(int p = 2; p*p <=n; p++) { // If prime[p] is not changed, then it is a prime if(prime[p] == true) { // Update all multiples of p for(int i = p*2; i <= n; i += p) prime[i] = false; } } // Print all prime numbers for(int i = 2; i <= n; i++) { if(prime[i] == true) System.out.print(i + " "); } } // Driver Program to test above function public static void main(String args[]) { int n = 30; System.out.print("Following are the prime numbers "); System.out.println("smaller than or equal to " + n); SieveOfEratosthenes g = new SieveOfEratosthenes(); g.sieveOfEratosthenes(n); } } // This code has been contributed by Amit Khandelwal.

## Python

# Python program to print all primes smaller than or equal to # n using Sieve of Eratosthenes def SieveOfEratosthenes(n): # Create a boolean array "prime[0..n]" and initialize # all entries it as true. A value in prime[i] will # finally be false if i is Not a prime, else true. prime = [True for i in range(n+1)] p = 2 while (p * p <= n): # If prime[p] is not changed, then it is a prime if (prime[p] == True): # Update all multiples of p for i in range(p * 2, n+1, p): prime[i] = False p += 1 # Print all prime numbers for p in range(2, n): if prime[p]: print p, # driver program if __name__=='__main__': n = 30 print "Following are the prime numbers smaller", print "than or equal to", n SieveOfEratosthenes(n)

Output:

Following are the prime numbers below 30 2 3 5 7 11 13 17 19 23 29

You may also like to see :

Segmented Sieve.

Sieve of Eratosthenes in 0(n) time complexity

References:

http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

This article is compiled by **Abhinav Priyadarshi** and reviewed by GeeksforGeeks team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above