input: 
  def sieve_of_eratosthenes(start: int, end: int) -> list:
      """
      Generates a list of prime numbers between two given numbers using the Sieve of Eratosthenes algorithm.
      >>> sieve_of_eratosthenes(1, 20)
      [2, 3, 5, 7, 11, 13, 17, 19]
      """
  
output:     if start < 2:
          start = 2
      is_prime = [True] * (end + 1)
      p = 2
      while (p * p <= end):
          if (is_prime[p] == True):
              for i in range(p * p, end + 1, p):
                  is_prime[i] = False
          p += 1
      prime_numbers = [p for p in range(start, end + 1) if is_prime[p]]
      return prime_numbers