Sieve of Eratosthenes

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Given a number n, find all prime numbers less than or equal to n.

Examples:

Input: n = 10
Output: [2, 3, 5, 7]
Explanation: The prime numbers up to 10 obtained by Sieve of Eratosthenes are [2, 3, 5, 7].

Input: n = 35
Output: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]
Explanation: The prime numbers up to 35 obtained by Sieve of Eratosthenes are [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31].

[Naive Approach] - Using Loop O(n*sqrt(n)) Time and O(1) Space

The Naive Approach for finding all prime numbers from 1 to n involves checking each number individually to determine whether it is prime.

Step By Step Implementations:

• Loop through all numbers i from 2 to n.
• For each i, check if it is divisible by any number from 2 to i - 1.
• If it is divisible, then i is not prime.
• If it is not divisible by any number in that range, then i is prime.

[Efficient Approach] - Sieve of Eratosthenes

The Sieve of Eratosthenes efficiently finds all primes up to n by repeatedly marking multiples of each prime as non-prime, starting from 2. This avoids redundant checks and quickly filters out all composite numbers.

Step By Step Implementations:

• Initialize a Boolean array prime[0..n] and set all entries to true, except for 0 and 1 (which are not primes).
• Start from 2, the smallest prime number.
• For each number p from 2 up to √n:
  • If p is marked as prime(true):
  • Mark all multiples of p as not prime(false), starting from p * p (since smaller multiples have already been marked by smaller primes).
• After the loop ends, all the remaining true entries in prime represent prime numbers.

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

vector<int> sieve(int n) {
    
    // creation of boolean array
    vector<bool> prime(n + 1, true);
    for (int p = 2; p * p <= n; p++) {
        if (prime[p] == true) {
            
            // marking as false
            for (int i = p * p; i <= n; i += p)
                prime[i] = false;
        }
    }
    
    vector<int> res;
    for (int p = 2; p <= n; p++){
        if (prime[p]){ 
            res.push_back(p);
        }
    }
    return res;
}

int main(){
    int n = 35;
    vector<int> res = sieve(n);
    for(auto ele : res){
        cout << ele << ' ';
    }
    return 0;
}

class GfG {
    static int[] sieve(int n) {
        
        // creation of boolean array
        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]) {
                // marking as false
                for (int i = p * p; i <= n; i += p)
                    prime[i] = false;
            }
        }
 
        // Count number of primes
        int count = 0;
        for (int p = 2; p <= n; p++) {
            if (prime[p])
                count++;
        }
 
        // Store primes in an array
        int[] res = new int[count];
        int index = 0;
        for (int p = 2; p <= n; p++) {
            if (prime[p])
                res[index++] = p;
        }
 
        return res;
    }
 
    public static void main(String[] args) {
        int n = 35;
        int[] res = sieve(n);
        for (int ele : res) {
            System.out.print(ele + " ");
        }
    }
}

def sieve(n):
   
    #Create a boolean list to track prime status of numbers
    prime = [True] * (n + 1)
    p = 2

    # Sieve of Eratosthenes algorithm
    while p * p <= n:
        if prime[p]:
            
            # Mark all multiples of p as non-prime
            for i in range(p * p, n + 1, p):
                prime[i] = False
        p += 1

    # Collect all prime numbers
    res = []
    for p in range(2, n + 1):
        if prime[p]:
            res.append(p)
    
    return res

if __name__ == "__main__":
    n = 35
    res = sieve(n)
    for ele in res:
        print(ele, end=' ')

using System;
using System.Collections.Generic;

class GfG {
    
    // Function to return all prime numbers up to n
    static List<int> sieve(int n)  {
        
        // Boolean array to mark primes
        bool[] prime = new bool[n + 1];
        for (int i = 0; i <= n; i++) {
            prime[i] = true;
        }

        // Sieve of Eratosthenes
        for (int p = 2; p * p <= n; p++)
        {
            if (prime[p])
            {
                for (int i = p * p; i <= n; i += p)
                {
                    prime[i] = false;
                }
            }
        }

        // Store primes in list
        List<int> res = new List<int>();
        for (int i = 2; i <= n; i++)
        {
            if (prime[i])
            {
                res.Add(i);
            }
        }

        return res;
    }

    static void Main()
    {
        int n = 35;
        List<int> res = sieve(n);

        foreach (int ele in res)
        {
            Console.Write(ele + " ");
        }
    }
}

function sieve(n) {
    
    // Create a boolean array to mark primes
    let prime = new Array(n + 1).fill(true);
    
    // 0 and 1 are not prime
    prime[0] = prime[1] = false; 

    // Apply Sieve of Eratosthenes
    for (let p = 2; p * p <= n; p++) {
        if (prime[p]) {
          
            // Mark all multiples of p as not prime
            for (let i = p * p; i <= n; i += p) {
                prime[i] = false;
            }
        }
    }

    // Collect all primes into result array
    let res = [];
    for (let p = 2; p <= n; p++) {
        if (prime[p]) {
            res.push(p);
        }
    }

    return res;
}

// Driver code
let n = 35;
let res = sieve(n);
console.log(res.join(' '));

2 3 5 7 11 13 17 19 23 29 31 

Time Complexity: O(n*log(log(n))). For each prime number, we mark its multiples, which takes around n/p steps. The total time is proportional to n*(1/2 + 1/3 + 1/5 + ....).
This sum over primes grows slowly and is approximately O(n*log(log(n))) making the algorithm very efficient.
Auxiliary Space: O(n)

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Sieve of Eratosthenes

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Sieve of Eratosthenes

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