Sieve of Eratosthenes
Last Updated :
06 Mar, 2024
Given a number n, print all primes smaller than or equal to n. It is also given that n is a small number.Â
Example:Â
Input : n =10
Output : 2 3 5 7Â
Input : n = 20Â
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.
Following is the algorithm to find all the prime numbers less than or equal to a given integer n by the Eratosthene’s method:Â
When the algorithm terminates, all the numbers in the list that are not marked are prime.
Explanation with Example:Â
Let us take an example when n = 100. So, we need to print all prime numbers smaller than or equal to 100.Â
We create a list of all numbers from 2 to 100. Â
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.Â
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. Â
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. Â
We move to our next unmarked number 7 and mark all multiples of 7 and are greater than or equal to the square of it. Â
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, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.
Implementation:Â
C++
#include <bits/stdc++.h>
using namespace std;
void SieveOfEratosthenes( int n)
{
bool prime[n + 1];
memset (prime, true , sizeof (prime));
for ( int p = 2; p * p <= n; p++) {
if (prime[p] == true ) {
for ( int i = p * p; i <= n; i += p)
prime[i] = false ;
}
}
for ( int p = 2; p <= n; p++)
if (prime[p])
cout << p << " " ;
}
int main()
{
int n = 30;
cout << "Following are the prime numbers smaller "
<< " than or equal to " << n << endl;
SieveOfEratosthenes(n);
return 0;
}
|
C
#include <stdio.h>
#include <stdbool.h>
#include <string.h>
void SieveOfEratosthenes( int n)
{
bool prime[n + 1];
memset (prime, true , sizeof (prime));
for ( int p = 2; p * p <= n; p++) {
if (prime[p] == true ) {
for ( int i = p * p; i <= n; i += p)
prime[i] = false ;
}
}
for ( int p = 2; p <= n; p++)
if (prime[p])
printf ( "%d " ,p);
}
int main()
{
int n = 30;
printf ( "Following are the prime numbers smaller than or equal to %d \n" , n);
SieveOfEratosthenes(n);
return 0;
}
|
Java
class SieveOfEratosthenes {
void sieveOfEratosthenes( int n)
{
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] == true ) {
for ( int i = p * p; i <= n; i += p)
prime[i] = false ;
}
}
for ( int i = 2 ; i <= n; i++) {
if (prime[i] == true )
System.out.print(i + " " );
}
}
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);
}
}
|
C#
using System;
namespace prime {
public class GFG {
public static void SieveOfEratosthenes( int n)
{
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] == true )
{
for ( int i = p * p; i <= n; i += p)
prime[i] = false ;
}
}
for ( int i = 2; i <= n; i++)
{
if (prime[i] == true )
Console.Write(i + " " );
}
}
public static void Main()
{
int n = 30;
Console.WriteLine(
"Following are the prime numbers" );
Console.WriteLine( "smaller than or equal to " + n);
SieveOfEratosthenes(n);
}
}
}
|
Javascript
<script>
function sieveOfEratosthenes(n)
{
prime = Array.from({length: n+1}, (_, i) => true );
for (p = 2; p * p <= n; p++)
{
if (prime[p] == true )
{
for (i = p * p; i <= n; i += p)
prime[i] = false ;
}
}
for (i = 2; i <= n; i++)
{
if (prime[i] == true )
document.write(i + " " );
}
}
var n = 30;
document.write(
"Following are the prime numbers " );
document.write( "smaller than or equal to " + n+ "<br>" );
sieveOfEratosthenes(n);
</script>
|
PHP
<?php
function SieveOfEratosthenes( $n )
{
$prime = array_fill (0, $n +1, true);
for ( $p = 2; $p * $p <= $n ; $p ++)
{
if ( $prime [ $p ] == true)
{
for ( $i = $p * $p ; $i <= $n ; $i += $p )
$prime [ $i ] = false;
}
}
for ( $p = 2; $p <= $n ; $p ++)
if ( $prime [ $p ])
echo $p . " " ;
}
$n = 30;
echo "Following are the prime numbers "
. "smaller than or equal to " . $n . "\n" ;
SieveOfEratosthenes( $n );
?>
|
Python3
def SieveOfEratosthenes(n):
prime = [ True for i in range (n + 1 )]
p = 2
while (p * p < = n):
if (prime[p] = = True ):
for i in range (p * p, n + 1 , p):
prime[i] = False
p + = 1
for p in range ( 2 , n + 1 ):
if prime[p]:
print (p)
if __name__ = = '__main__' :
n = 20
print ( "Following are the prime numbers smaller" ),
print ( "than or equal to" , n)
SieveOfEratosthenes(n)
|
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)))
Auxiliary Space: O(N)
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