Sieve of Eratosthenes

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

Example: 

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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.  

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. 

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.  

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.  

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 the 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;
}

C

// C program to print all primes smaller than or equal to
// n using Sieve of Eratosthenes
#include <stdio.h>
#include <stdbool.h>
#include <string.h>

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])
            printf("%d ",p);
}

// Driver Code
int main()
{
    int n = 30;
    printf("Following are the prime numbers smaller than or equal to %d \n", n);
    SieveOfEratosthenes(n);
    return 0;
}

// This code is contributed by Aditya Kumar (adityakumar129)

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 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 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 is contributed by Aditya Kumar (adityakumar129)

Python3

# 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 = 20
    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
?>

Javascript

<script>

// javascript 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.from({length: n+1}, (_, i) => 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 (i = 2; i <= n; i++)
    {
        if (prime[i] == true)
            document.write(i + " ");
    }
}

// Driver Code
var n = 30;
document.write(
    "Following are the prime numbers ");
document.write("smaller than or equal to " + n+"<br>");
sieveOfEratosthenes(n);

// This code is contributed by 29AjayKumar

</script>

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

Auxiliary Space: O(n)

C++

// the following implementation
// stores only halves of odd numbers
// the algorithm is a faster by some constant factors

#include <bitset>
#include <iostream>
using namespace std;

bitset<500001> Primes;
void SieveOfEratosthenes(int n)
{
    Primes[0] = 1;
    for (int i = 3; i*i <= n; i += 2) {
        if (Primes[i / 2] == 0) {
            for (int j = 3 * i; j <= n; j += 2 * i)
                Primes[j / 2] = 1;
        }
    }
}

int main()
{
    int n = 100;
    SieveOfEratosthenes(n);
    for (int i = 1; i <= n; i++) {
        if (i == 2)
            cout << i << ' ';
        else if (i % 2 == 1 && Primes[i / 2] == 0)
            cout << i << ' ';
    }
    return 0;
}

Java

// Java program for the above approach
import java.io.*;
public class GFG {

  static int[] Primes = new int[500001];

  static void SieveOfEratosthenes(int n)
  {
    Primes[0] = 1;
    for (int i = 3; i * i <= n; i += 2) {
      if (Primes[i / 2] == 0) {
        for (int j = 3 * i; j <= n; j += 2 * i)
          Primes[j / 2] = 1;
      }
    }
  }

  // Driver Code
  public static void main(String[] args)
  {

    int n = 100;
    SieveOfEratosthenes(n);
    for (int i = 1; i <= n; i++) {
      if (i == 2)
        System.out.print(i + " ");
      else if (i % 2 == 1 && Primes[i / 2] == 0)
        System.out.print(i + " ");
    }
  }
}

// This code is contributed by ukasp.

Python3

# Python program for the above approach
Primes = [0] * 500001
def SieveOfEratosthenes(n) :

    Primes[0] = 1
    i = 3
    while(i*i <= n) :
        if (Primes[i // 2] == 0) :
            for j in range(3 * i, n+1, 2 * i) :
                Primes[j // 2] = 1

        i += 2

# Driver Code
if __name__ == "__main__":

    n = 100
    SieveOfEratosthenes(n)
    for i in range(1, n+1) :
        if (i == 2) :
            print( i, end = " ")
        elif (i % 2 == 1 and Primes[i // 2] == 0) :
            print( i, end = " ")

    # This code is contributed by code_hunt.

C#

// C# program for the above approach
using System;
public class GFG {

  static int[] Primes = new int[500001];

  static void SieveOfEratosthenes(int n)
  {
    Primes[0] = 1;
    for (int i = 3; i*i <= n; i += 2) {
      if (Primes[i / 2] == 0) {
        for (int j = 3 * i; j <= n; j += 2 * i)
          Primes[j / 2] = 1;
      }
    }
  }

  // Driver Code
  public static void Main(String[] args) {

    int n = 100;
    SieveOfEratosthenes(n);
    for (int i = 1; i <= n; i++) {
      if (i == 2)
        Console.Write(i + " ");
      else if (i % 2 == 1 && Primes[i / 2] == 0)
        Console.Write(i + " ");
    }
  }
}

// This code is contributed by sanjoy_62.

Javascript

// A JavaScript Program
// the following implementation
// stores only halves of odd numbers
// the algorithm is a faster by some constant factors

let Primes = new Array(500001).fill(0);

function SieveOfEratosthenes(n)
{
    Primes[0] = 1;
    for (let i = 3; i*i <= n; i += 2) {
        if (Primes[Math.floor(i / 2)] == 0) {
            for (let j = 3 * i; j <= n; j += 2 * i){
                 Primes[Math.floor(j / 2)] = 1;
            }
        }
    }
}

let n = 100;
SieveOfEratosthenes(n);
let res = "";
for (let i = 1; i <= n; i++) {
    if (i == 2){
        res = res + i + " ";
    }
    else if (i % 2 == 1 && Primes[Math.floor(i / 2)] == 0){
        res = res + i + " ";
    }
}
console.log(res);

// The code is contributed by Gautam goel (gautamgoel962)

You may also like to see :  

• How is the time complexity of Sieve of Eratosthenes is n*log(log(n))?
• Segmented Sieve.
• Sieve of Eratosthenes in 0(n) time complexity

Auxiliary Space: O(1)