input: def sieve_of_eratosthenes(start, end):
      """ Generate 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:     primes = [True] * (end + 1)
      primes[0] = primes[1] = False
      for i in range(2, int(end**0.5) + 1):
          if primes[i]:
              for j in range(i*i, end + 1, i):
                  primes[j] = False
      return [num for num in range(start, end + 1) if primes[num]]