Sieve of Eratosthenes

Write a C++ program to implement the Sieve of Eratosthenes algorithm to find all prime numbers up to a given number `n`.

Example 1

Input: n = 10
Output: 2 3 5 7
Explanation:
   

   
Step 1: Create a boolean array of size n+1 initialized as true (assuming all numbers are prime initially).
   
Step 2: Mark 0 and 1 as non-prime (false).
   
Step 3: Starting from 2, iterate through each number. If it's marked as prime, mark all its multiples as non-prime.
   
Step 4: For n = 10, after sieving:
      

      
2 is prime, mark 4, 6, 8, 10 as non-prime
      
3 is prime, mark 6, 9 as non-prime
      
4 is already marked as non-prime
      
5 is prime, mark 10 as non-prime
      
   
   
Step 5: Return all numbers still marked as prime: 2, 3, 5, 7.
   

Example 2

Input: n = 20
Output: 2 3 5 7 11 13 17 19
Explanation:
   

   
Step 1: Create a boolean array of size n+1 initialized as true.
   
Step 2: Mark 0 and 1 as non-prime.
   
Step 3: Starting from 2, iterate through each number. If it's marked as prime, mark all its multiples as non-prime.
   
Step 4: For n = 20, after sieving:
      

      
2 is prime, mark 4, 6, 8, 10, 12, 14, 16, 18, 20 as non-prime
      
3 is prime, mark 6, 9, 12, 15, 18 as non-prime
      
5 is prime, mark 10, 15, 20 as non-prime
      
7 is prime, mark 14 as non-prime
      
   
   
Step 5: Return all numbers still marked as prime: 2, 3, 5, 7, 11, 13, 17, 19.
   

Constraints

1 ≤ n ≤ 10^6
Time Complexity: O(n log log n)
Space Complexity: O(n)