"""simple docstring"""






import math



def 						a_		(      lowerCamelCase ):

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

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

								UpperCAmelCase__    =							[3, 5]
								UpperCAmelCase__    =							2
								UpperCAmelCase__    =							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):
												lowerCAmelCase__						:						Tuple		  =			0
												try:
																		lowerCAmelCase__						:						Any		  =			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"""






def 						a_		(      lowerCamelCase      ,    lowerCamelCase ):
				return x if y == 0 else greatest_common_divisor(lowerCamelCase      ,    x % y )



def 						a_		(      lowerCamelCase      ,    lowerCamelCase ):
				return (x * y) // greatest_common_divisor(lowerCamelCase      ,    lowerCamelCase )



def 						a_		(      lowerCamelCase = 2_0 ):
				UpperCAmelCase__    =							1
				for i in range(1      ,    n + 1 ):
								UpperCAmelCase__    =							lcm(lowerCamelCase      ,    lowerCamelCase )
				return g


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



