from __future__ import annotations

import math





def  _lowerCamelCase					(   a_				: int):

			if num <= 0:
						lowerCamelCase						:Union[str, Any]						  =     F"{num}: Invalid input, please enter a positive integer."
						raise ValueError(a_)

			lowerCamelCase						:Any						  =     [True] * (num + 1)
			lowerCamelCase						:Union[str, Any]						  =     []
			lowerCamelCase						:List[Any]						  =     2
			lowerCamelCase						:Dict						  =     int(math.sqrt(a_))

			while start <= end:
						# If start is a prime
						if sieve[start] is True:
									prime.append(a_)

									# Set multiples of start be False
									for i in range(start * start						,			num + 1						,			a_):
												if sieve[i] is True:
															lowerCamelCase						:Optional[int]						  =     False

						start += 1

			for j in range(end + 1						,			num + 1):
						if sieve[j] is True:
									prime.append(a_)

			return prime


if __name__ == "__main__":
			print(prime_sieve(int(input("""Enter a positive integer: """).strip())))




import math





def  _lowerCamelCase					(   a_				: int):

			if 1 < number < 4:
						# 2 and 3 are primes
						return True
			elif number < 2 or number % 2 == 0 or number % 3 == 0:
						# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
						return False

			# All primes number are in format of 6k +/- 1
			for i in range(5						,			int(math.sqrt(a_) + 1)						,			6):
						if number % i == 0 or number % (i + 2) == 0:
									return False
			return True





def  _lowerCamelCase					(   a_				: float = 0.1):
			lowerCamelCase						:Dict						  =     3
			lowerCamelCase						:List[Any]						  =     3

			while primes / (2 * j - 1) >= ratio:
						for i in range(j * j + j + 1						,			(j + 2) * (j + 2)						,			j + 1):
									primes += is_prime(a_)
						j += 2
			return j


if __name__ == "__main__":
			import doctest

			doctest.testmod()



