



from math import isqrt






def 			lowerCAmelCase__	(					lowerCamelCase_	:	int):





				'''simple docstring'''

				lowerCAmelCase__  :       Tuple				=  [True] * max_number
				for i in range(2 ,isqrt(max_number - 1) + 1):
								if is_prime[i]:
												for j in range(i**2 ,lowerCamelCase_ ,lowerCamelCase_):
																lowerCAmelCase__  :       Optional[Any]				=  False

				return [i for i in range(2 ,lowerCamelCase_) if is_prime[i]]






def 			lowerCAmelCase__	(					lowerCamelCase_	:	int = 10**8):





				'''simple docstring'''

				lowerCAmelCase__  :       Optional[Any]				=  calculate_prime_numbers(max_number // 2)

				lowerCAmelCase__  :       List[str]				=  0
				lowerCAmelCase__  :       List[str]				=  0
				lowerCAmelCase__  :       Dict				=  len(lowerCamelCase_) - 1
				while left <= right:
								while prime_numbers[left] * prime_numbers[right] >= max_number:
												right -= 1
								semiprimes_count += right - left + 1
								left += 1

				return semiprimes_count


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






def 			lowerCAmelCase__	(					lowerCamelCase_	:	int):





				'''simple docstring'''

				if length <= 0 or not isinstance(lowerCamelCase_ ,lowerCamelCase_):
								raise ValueError('''Length must be a positive integer.''')
				return [n * (2 * n - 1) for n in range(lowerCamelCase_)]


if __name__ == "__main__":
							print(hexagonal_numbers(length=5))
							print(hexagonal_numbers(length=1_0))

