



def  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


				if num <= 0:
								raise ValueError("Input must be a positive integer"						)

				lowercase__					:   List[str]													=					[True] * (num + 1)

				lowercase__					:   Union[str, Any]													=					2
				while p * p <= num:
								if primes[p]:
												for i in range(p * p					,			num + 1					,			lowerCamelCase__						):
																lowercase__					:   Any													=					False
								p += 1

				return [prime for prime in range(2					,			num + 1						) if primes[prime]]


if __name__ == "__main__":
				import doctest

				doctest.testmod()

				lowerCAmelCase__					  =  int(input('''Enter a positive integer: ''').strip())
				print(prime_sieve_eratosthenes(user_num))









from math import ceil, sqrt


def  __lowerCamelCase (       lowerCamelCase__ = 1_000_000						):

				"""simple docstring"""


				lowercase__					:   int													=					0

				for outer_width in range(3					,			(limit // 4) + 2						):
								if outer_width**2 > limit:
												lowercase__					:   List[str]													=					max(ceil(sqrt(outer_width**2 - limit						)						)					,			1						)
								else:
												lowercase__					:   List[str]													=					1
								if (outer_width - hole_width_lower_bound) % 2:
												hole_width_lower_bound += 1

								answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1

				return answer


if __name__ == "__main__":
				print(f'''{solution() = }''')




