





'''simple docstring'''


def 		__a(SCREAMING_SNAKE_CASE_     :    list[int]			):



    '''simple docstring'''





    if not numbers:
        return 0

    if not isinstance(UpperCAmelCase__						,				(list, tuple)			) or not all(
        isinstance(UpperCAmelCase__						,				UpperCAmelCase__			) for number in numbers			):
        raise ValueError("numbers must be an iterable of integers"			)

    _lowerCAmelCase			=					numbers[0]

    for i in range(1						,				len(UpperCAmelCase__			)			):
        # update the maximum and minimum subarray products
        _lowerCAmelCase			=					numbers[i]
        if number < 0:
            _lowerCAmelCase			=					min_till_now, max_till_now
        _lowerCAmelCase			=					max(UpperCAmelCase__						,				max_till_now * number			)
        _lowerCAmelCase			=					min(UpperCAmelCase__						,				min_till_now * number			)

        # update the maximum product found till now
        _lowerCAmelCase			=					max(UpperCAmelCase__						,				UpperCAmelCase__			)

    return max_prod



'''simple docstring'''




import math




def 							lowerCamelCase	(      UpperCAmelCase__       :					int )			->      int:

							if not isinstance(UpperCAmelCase__      ,      UpperCAmelCase__ ):
														lowercase_					:			List[Any]										=       F'''Input value of [number={number}] must be an integer'''
														raise TypeError(UpperCAmelCase__ )

							if number < 1:
														lowercase_					:			Tuple										=       F'''Input value of [number={number}] must be > 0'''
														raise ValueError(UpperCAmelCase__ )
							elif number == 1:
														return 3
							elif number == 2:
														return 5
							else:
														lowercase_					:			Union[str, Any]										=       int(math.log(number // 3      ,      2 ) ) + 2

														lowercase_					:			Any										=       [3, 5]
														lowercase_					:			List[Any]										=       2
														lowercase_					:			Optional[Any]										=       3
														for block in range(1      ,      UpperCAmelCase__ ):
																					for _ in range(UpperCAmelCase__ ):
																												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):
												_lowercase						: Optional[Any]						=						0
												try:
																		_lowercase						: str						=						proth(number)
												except ValueError:
																		print(f"""ValueError: there is no {number}th Proth number""")
																		continue

												print(f"""The {number}th Proth number: {value}""")





