





import math

def    A__(		__lowerCAmelCase			):






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

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

								_snake_case			:      Union[str, Any]	    = [3, 5]
								_snake_case			:      Optional[Any]	    = 2
								_snake_case			:      Optional[int]	    = 3
								for block in range(1						,      lowerCAmelCase_			):
												for _ in range(lowerCAmelCase_			):
																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_		:					Any      						=    0
				try:
						lowercase_		:					int      						=    proth(number)
				except ValueError:
						print(F'''ValueError: there is no {number}th Proth number''')
						continue

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










def       _lowerCAmelCase    (				lowerCAmelCase_				:int		,    lowerCAmelCase_				:int				)->str:
   '''simple docstring'''






   if not isinstance(lowerCAmelCase_		,    lowerCAmelCase_				):
      raise ValueError("iterations must be defined as integers"				)
   if not isinstance(lowerCAmelCase_		,    lowerCAmelCase_				) or not number >= 1:
      raise ValueError(
          "starting number must be\n                         and integer and be more than 0"				)
   if not iterations >= 1:
      raise ValueError("Iterations must be done more than 0 times to play FizzBuzz"				)

   snake_case_            =       ""
   while number <= iterations:
      if number % 3 == 0:
         out += "Fizz"
      if number % 5 == 0:
         out += "Buzz"
      if 0 not in (number % 3, number % 5):
         out += str(lowerCAmelCase_				)

      # print(out)
      number += 1
      out += " "
   return out


if __name__ == "__main__":
   import doctest

   doctest.testmod()




