





"""simple docstring"""






def     __A      (			a_						:int)		->		list[int]:

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


if __name__ == "__main__":
		print(hexagonal_numbers(length=5))
		print(hexagonal_numbers(length=10))






"""simple docstring"""






from math import isqrt, loga

def     __A      (			a_						:int)		->		list[int]:
				__a :						int			       =  [True] * max_number
				for i in range(2   ,       isqrt(max_number - 1) + 1):
								if is_prime[i]:
												for j in range(i**2   ,       a_   ,       a_):
																__a :						int			       =  False

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

def     __A      (			a_						:int = 80_08_00   ,       a_						:int = 80_08_00)		->		int:
				__a :						str			       =  degree * loga(a_)
				__a :						Tuple			       =  int(a_)
				__a :						int			       =  calculate_prime_numbers(a_)

				__a :						List[Any]			       =  0
				__a :						Optional[Any]			       =  0
				__a :						Dict			       =  len(a_) - 1
				while left < right:
								while (
								    prime_numbers[right] * loga(prime_numbers[left])
								    + prime_numbers[left] * loga(prime_numbers[right])
								    > upper_bound
								):
												right -= 1
								hybrid_integers_count += right - left
								left += 1

				return hybrid_integers_count


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