input: def sieve_eratosthenes(n): 
      """ Find all prime numbers up to a given limit n using the Sieve of Eratosthenes algorithm.
      >>> sieve_eratosthenes(10)
      [2, 3, 5, 7]
      >>> sieve_eratosthenes(20)
      [2, 3, 5, 7, 11, 13, 17, 19]
      """
      prime_list = [True] * (n+1) 
      prime_list[0] = False
      prime_list[1] = False
      primes = [] 
        
      for i in range(2, n+1): 
          if prime_list[i] == True: 
              primes.append(i) 
              for j in range(i*i, n+1, i): 
                  prime_list[j] = False
      return primes
output:     prime_list = [True] * (n+1) 
      prime_list[0] = False
      prime_list[1] = False
      primes = [] 
        
      for i in range(2, n+1): 
          if prime_list[i] == True: 
              primes.append(i) 
              for j in range(i*i, n+1, i): 
                  prime_list[j] = False
      return primes