





'''simple docstring'''






import math






def      __A						(							a_						:  int	):

		if not isinstance(a_      ,a_	):
				lowerCAmelCase     :    str						=   f'''Input value of [number={number}] must be an integer'''
				raise TypeError(a_	)

		if number < 1:
				lowerCAmelCase     :    Optional[Any]						=   f'''Input value of [number={number}] must be > 0'''
				raise ValueError(a_	)
		elif number == 1:
				return 3
		elif number == 2:
				return 5
		else:
				lowerCAmelCase     :    Optional[int]						=   int(math.log(number // 3      ,2	)	) + 2

				lowerCAmelCase     :    List[Any]						=   [3, 5]
				lowerCAmelCase     :    Optional[Any]						=   2
				lowerCAmelCase     :    Union[str, Any]						=   3
				for block in range(1      ,a_	):
						for _ in range(a_	):
								proth_list.append(2 ** (block + 1) + proth_list[proth_index - 1]	)
								proth_index += 1
						increment *= 2

		return proth_list[number - 1]


if __name__ == "__main__":
					import doctest

					doctest.testmod()

					for number in range(11):
										lowerCAmelCase       		=      0
										try:
															lowerCAmelCase       		=      proth(number)
										except ValueError:
															print(F'''ValueError: there is no {number}th Proth number''')
															continue

										print(F'''The {number}th Proth number: {value}''')










'''simple docstring'''






import numpy as np






def      __A						(							a_						:  np.array	):
		return 1 / (1 + np.exp(-vector	))


if __name__ == "__main__":
					import doctest

					doctest.testmod()



