Sieve of Eratosthenes

Difficulty Level : Medium
Last Updated : 20 Jan, 2021

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

Example:

Input : n =10
Output : 2 3 5 7

Input : n = 20
Output: 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 the Eratosthene’s method:
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 prime numbers smaller than or equal to 50.
We create a list of all numbers from 2 to 50.

Sieve1

According to the algorithm we will mark all the numbers which are divisible by 2 and are greater than or equal to the square of it.

sieve2

Now we move to our next unmarked number 3 and mark all the numbers which are multiples of 3 and are greater than or equal to the square of it.

SieveofEratosthenes3

We move to our next unmarked number 5 and mark all multiples of 5 and are greater than or equal to the square of it.

Sieve4

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

Sieve5

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 the 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++ 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 greater than or
            // equal to the square of it
            // numbers which are multiple 
            // of p and are less than p^2 
            // are already been marked.
            for (int i = p * p; i <= n; i += p)
                prime[i] = false;
        }
    }
 
    // Print all prime numbers
    for (int p = 2; p <= n; p++)
        if (prime[p])
            cout << p << " ";
}
 
// Driver Code
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 * p; 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 Code
    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 * p, n+1, p):
                prime[i] = False
        p += 1
 
    # Print all prime numbers
    for p in range(2, n+1):
        if prime[p]:
            print p,
 
 
# Driver code
if __name__ == '__main__':
    n = 30
    print "Following are the prime numbers smaller",
    print "than or equal to", n
    SieveOfEratosthenes(n)

C#

// C# program to print all primes
// smaller than or equal to n
// using Sieve of Eratosthenes
using System;
 
namespace prime {
public class GFG {
 
    public static 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 = new bool[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 * p; i <= n; i += p)
                    prime[i] = false;
            }
        }
 
        // Print all prime numbers
        for (int i = 2; i <= n; i++)
        {
            if (prime[i] == true)
                Console.Write(i + " ");
        }
    }
 
    // Driver Code
    public static void Main()
    {
        int n = 30;
        Console.WriteLine(
            "Following are the prime numbers");
        Console.WriteLine("smaller than or equal to " + n);
        SieveOfEratosthenes(n);
    }
}
}
 
// This code is contributed by Sam007.

PHP

<?php
// php program to print all primes smaller
// than or equal to n using Sieve of
// Eratosthenes
 
function 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 = array_fill(0, $n+1, true);
 
    for ($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 ($i = $p*$p; $i <= $n; $i += $p)
                $prime[$i] = false;
        }
    }
 
    // Print all prime numbers
    for ($p = 2; $p <= $n; $p++)
        if ($prime[$p])
            echo $p." ";
}
 
// Driver Code
    $n = 30;
    echo "Following are the prime numbers "
     ."smaller than or equal to " .$n."\n" ;
    SieveOfEratosthenes($n);
 
// This code is contributed by mits
?>

Output

Following are the prime numbers smaller  than or equal to 30
2 3 5 7 11 13 17 19 23 29

Time complexity : O(n*log(log(n)))

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