






import math
from collections.abc import Generator



def      _SCREAMING_SNAKE_CASE  (	__lowercase				:     int      )  -> str:





     """simple docstring"""

     __A             =		(i for i in range(1     ,			(max_n + 1)      ))
     for i in (n for n in numbers if n > 1):
          for j in range(2     ,			__lowercase      ):
               if (i % j) == 0:
                    break
          else:
               yield i



def      _SCREAMING_SNAKE_CASE  (	__lowercase				:     int      )  -> Any:





     """simple docstring"""

     __A             =		(i for i in range(1     ,			(max_n + 1)      ))
     for i in (n for n in numbers if n > 1):
          # only need to check for factors up to sqrt(i)
          __A             =		int(math.sqrt(__lowercase      )      ) + 1
          for j in range(2     ,			__lowercase      ):
               if (i % j) == 0:
                    break
          else:
               yield i



def      _SCREAMING_SNAKE_CASE  (	__lowercase				:     int      )  -> List[str]:





     """simple docstring"""

     __A             =		(i for i in range(1     ,			(max_n + 1)     ,			2      ))
     # It's useless to test even numbers as they will not be prime
     if max_n > 2:
          yield 2  # Because 2 will not be tested, it's necessary to yield it now
     for i in (n for n in numbers if n > 1):
          __A             =		int(math.sqrt(__lowercase      )      ) + 1
          for j in range(3     ,			__lowercase     ,			2      ):
               # As we removed the even numbers, we don't need them now
               if (i % j) == 0:
                    break
          else:
               yield i



def      _SCREAMING_SNAKE_CASE  (	)  -> List[str]:





     """simple docstring"""

     from timeit import timeit

     __A             =		"from __main__ import slow_primes, primes, fast_primes"
     print(timeit("""slow_primes(1_000_000_000_000)"""     ,			setup=__lowercase     ,			number=1_0_0_0_0_0_0      )      )
     print(timeit("""primes(1_000_000_000_000)"""     ,			setup=__lowercase     ,			number=1_0_0_0_0_0_0      )      )
     print(timeit("""fast_primes(1_000_000_000_000)"""     ,			setup=__lowercase     ,			number=1_0_0_0_0_0_0      )      )


if __name__ == "__main__":
  __a :    Any								=				int(input("Calculate primes up to:\n>> ").strip())
  for ret in primes(number):
    print(ret)
  benchmark()












def      _SCREAMING_SNAKE_CASE  (	__lowercase				:     list[int]      )  -> list[int]:





    """simple docstring"""

    if len(__lowercase      ) > 1:
        __A             =		len(__lowercase      ) // 2  # Finds the middle of the array
        __A             =		arr[
            :middle_length
        ]  # Creates an array of the elements in the first half.
        __A             =		arr[
            middle_length:
        ]  # Creates an array of the elements in the second half.
        __A             =		len(__lowercase      )
        __A             =		len(__lowercase      )
        merge(__lowercase      )  # Starts sorting the left.
        merge(__lowercase      )  # Starts sorting the right
        __A             =		0  # Left Counter
        __A             =		0  # Right Counter
        __A             =		0  # Position Counter
        while (
            left_index < left_size and right_index < right_size
        ):  # Runs until the lowers size of the left and right are sorted.
            if left_array[left_index] < right_array[right_index]:
                __A             =		left_array[left_index]
                left_index += 1
            else:
                __A             =		right_array[right_index]
                right_index += 1
            index += 1
        while (
            left_index < left_size
        ):  # Adds the left over elements in the left half of the array
            __A             =		left_array[left_index]
            left_index += 1
            index += 1
        while (
            right_index < right_size
        ):  # Adds the left over elements in the right half of the array
            __A             =		right_array[right_index]
            right_index += 1
            index += 1
    return arr


if __name__ == "__main__":
  import doctest

  doctest.testmod()




