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:
Input : n =10
Output : 2 3 5 7Input : 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.
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 = 100. So, we need to print all prime numbers smaller than or equal to 100.
We create a list of all numbers from 2 to 100.
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 move to our next unmarked number 7 and mark all multiples of 7 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, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.
Implementation:
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) |
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. |
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> |
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 ?> |
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) |
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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