# Sieve of Eratosthenes in 0(n) time complexity

• Difficulty Level : Hard
• Last Updated : 14 Jul, 2022

The classical Sieve of Eratosthenes algorithm takes O(N log (log N)) time to find all prime numbers less than N. In this article, a modified Sieve is discussed that works in O(N) time.
Example :

```Given a number N, print all prime
numbers smaller than N

Input :  int N = 15
Output : 2 3 5 7 11 13

Input : int N = 20
Output : 2 3 5 7 11 13 17 19```

Manipulated Sieve of Eratosthenes algorithm works as follows:

```For every number i where i varies from 2 to N-1:
Check if the number is prime. If the number
is prime, store it in prime array.

For every prime numbers j less than or equal to the smallest
prime factor p of i:
Mark all numbers i*p as non_prime.
Mark smallest prime factor of i*p as j```

Below is the implementation of the above idea.

## C++

 `// C++ program to generate all prime numbers``// less than N in O(N)``#include``using` `namespace` `std;``const` `long` `long` `MAX_SIZE = 1000001;` `// isPrime[] : isPrime[i] is true if number is prime``// prime[] : stores all prime number less than N``// SPF[] that store smallest prime factor of number``// [for Exp : smallest prime factor of '8' and '16'``// is '2' so we put SPF[8] = 2 , SPF[16] = 2 ]``vector<``long` `long` `>isprime(MAX_SIZE , ``true``);``vector<``long` `long` `>prime;``vector<``long` `long` `>SPF(MAX_SIZE);` `// function generate all prime number less than N in O(n)``void` `manipulated_seive(``int` `N)``{``    ``// 0 and 1 are not prime``    ``isprime[0] = isprime[1] = ``false` `;` `    ``// Fill rest of the entries``    ``for` `(``long` `long` `int` `i=2; i

## Java

 `// Java program to generate all prime numbers``// less than N in O(N)`  `import` `java.util.Vector;` `class` `Test``{``    ``static` `final` `int` `MAX_SIZE = ``1000001``;``    ``// isPrime[] : isPrime[i] is true if number is prime``    ``// prime[] : stores all prime number less than N``    ``// SPF[] that store smallest prime factor of number``    ``// [for Exp : smallest prime factor of '8' and '16'``    ``// is '2' so we put SPF[8] = 2 , SPF[16] = 2 ]``    ``static` `Vectorisprime = ``new` `Vector<>(MAX_SIZE);``    ``static` `Vectorprime = ``new` `Vector<>();``    ``static` `VectorSPF = ``new` `Vector<>(MAX_SIZE);``     ` `    ``// method generate all prime number less than N in O(n)``    ``static` `void` `manipulated_seive(``int` `N)``    ``{``        ``// 0 and 1 are not prime``        ``isprime.set(``0``, ``false``);``        ``isprime.set(``1``, ``false``);``        ` `        ``// Fill rest of the entries``        ``for` `(``int` `i=``2``; i

## Python3

 `# Python3 program to generate all``# prime numbers less than N in O(N)` `MAX_SIZE ``=` `1000001` `# isPrime[] : isPrime[i] is true if``#             number is prime``# prime[] : stores all prime number``#           less than N``# SPF[] that store smallest prime``# factor of number [for ex : smallest``# prime factor of '8' and '16'``# is '2' so we put SPF[8] = 2 ,``# SPF[16] = 2 ]``isprime ``=` `[``True``] ``*` `MAX_SIZE``prime ``=` `[]``SPF ``=` `[``None``] ``*` `(MAX_SIZE)` `# function generate all prime number``# less than N in O(n)``def` `manipulated_seive(N):` `    ``# 0 and 1 are not prime``    ``isprime[``0``] ``=` `isprime[``1``] ``=` `False` `    ``# Fill rest of the entries``    ``for` `i ``in` `range``(``2``, N):``    ` `        ``# If isPrime[i] == True then i is``        ``# prime number``        ``if` `isprime[i] ``=``=` `True``:``        ` `            ``# put i into prime[] vector``            ``prime.append(i)` `            ``# A prime number is its own smallest``            ``# prime factor``            ``SPF[i] ``=` `i``        ` `        ``# Remove all multiples of i*prime[j]``        ``# which are not prime by making is``        ``# Prime[i * prime[j]] = false and put``        ``# smallest prime factor of i*Prime[j]``        ``# as prime[j] [ for exp :let i = 5 , j = 0 ,``        ``# prime[j] = 2 [ i*prime[j] = 10 ]``        ``# so smallest prime factor of '10' is '2'``        ``# that is prime[j] ] this loop run only one``        ``# time for number which are not prime``        ``j ``=` `0``        ``while` `(j < ``len``(prime) ``and``               ``i ``*` `prime[j] < N ``and``                   ``prime[j] <``=` `SPF[i]):``        ` `            ``isprime[i ``*` `prime[j]] ``=` `False` `            ``# put smallest prime factor of i*prime[j]``            ``SPF[i ``*` `prime[j]] ``=` `prime[j]``            ` `            ``j ``+``=` `1``        ` `# Driver Code``if` `__name__ ``=``=` `"__main__"``:` `    ``N ``=` `13` `# Must be less than MAX_SIZE` `    ``manipulated_seive(N)` `    ``# print all prime number less than N``    ``i ``=` `0``    ``while` `i < ``len``(prime) ``and` `prime[i] <``=` `N:``        ``print``(prime[i], end ``=` `" "``)``        ``i ``+``=` `1``        ` `# This code is contributed by Rituraj Jain`

## C#

 `// C# program to generate all prime numbers``// less than N in O(N)``using` `System;``using` `System.Collections.Generic;` `class` `Test {``  ``static` `int` `MAX_SIZE = 1000001;` `  ``// isPrime[] : isPrime[i] is true if number is prime``  ``// prime[] : stores all prime number less than N``  ``// SPF[] that store smallest prime factor of number``  ``// [for Exp : smallest prime factor of '8' and '16'``  ``// is '2' so we put SPF[8] = 2 , SPF[16] = 2 ]``  ``static` `List<``bool``> isprime = ``new` `List<``bool``>(MAX_SIZE);``  ``static` `List<``int``> prime = ``new` `List<``int``>();``  ``static` `List<``int``> SPF = ``new` `List<``int``>(MAX_SIZE);` `  ``// method generate all prime number less than N in O(n)``  ``static` `void` `manipulated_seive(``int` `N)``  ``{``    ``// 0 and 1 are not prime``    ``isprime[0] = ``false``;``    ``isprime[1] = ``false``;` `    ``// Fill rest of the entries``    ``for` `(``int` `i = 2; i < N; i++)``    ``{` `      ``// If isPrime[i] == True then i is``      ``// prime number``      ``if` `(isprime[i])``      ``{` `        ``// put i into prime[] vector``        ``prime.Add(i);` `        ``// A prime number is its own smallest``        ``// prime factor``        ``SPF[i] = i;``      ``}` `      ``// Remove all multiples of  i*prime[j] which are``      ``// not prime by making isPrime[i*prime[j]] =``      ``// false and put smallest prime factor of``      ``// i*Prime[j] as prime[j] [for exp :let  i = 5,``      ``// j = 0, prime[j] = 2 [ i*prime[j] = 10] so``      ``// smallest prime factor of '10' is '2' that is``      ``// prime[j] ] this loop run only one time for``      ``// number which are not prime``      ``for` `(``int` `j = 0;``           ``j < prime.Count && i * prime[j] < N``           ``&& prime[j] <= SPF[i];``           ``j++) {``        ``isprime[i * prime[j]] = ``false``;` `        ``// put smallest prime factor of i*prime[j]``        ``SPF[i * prime[j]] = prime[j];``      ``}``    ``}``  ``}` `  ``// Driver method``  ``public` `static` `void` `Main(``string``[] args)``  ``{``    ``int` `N = 13; ``// Must be less than MAX_SIZE` `    ``// initializing isprime and spf``    ``for` `(``int` `i = 0; i < MAX_SIZE; i++) {``      ``isprime.Add(``true``);``      ``SPF.Add(2);``    ``}` `    ``manipulated_seive(N);` `    ``// print all prime number less than N``    ``for` `(``int` `i = 0; i < prime.Count && prime[i] <= N;``         ``i++)``      ``Console.Write(prime[i] + ``" "``);``  ``}``}` `// This code is contributed by phasing17`

## PHP

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## Javascript

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Output :

`2 3 5 7 11`

Auxiliary Space: O(1)
Illustration:

```isPrime[0] = isPrime[1] = 0

After i = 2 iteration :
isPrime[]   [F, F, T, T, F, T, T, T]
SPF[]       [0, 0, 2, 0, 2, 0, 0, 0]
index   0  1  2  3  4  5  6  7

After i = 3 iteration :
isPrime[]  [F, F, T, T, F, T, F, T, T, F ]
SPF[]      [0, 0, 2, 3, 2, 0, 2, 0, 0, 3 ]
index     0  1  2  3  4  5  6  7  8  9

After i = 4 iteration :
isPrime[]  [F, F, T, T, F, T, F, T, F, F]
SPF[]      [0, 0, 2, 3, 2, 0, 2, 0, 2, 3]
index     0  1  2  3  4  5  6  7  8  9```

This article is contributed by Divyanshu Srivastava and Nishant Singh. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.