"""simple docstring"""







import math


def      __lowerCAmelCase(		__UpperCAmelCase			):






	"""simple docstring"""





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

		_lowercase :    Union[str, Any]							=							[3, 5]
		_lowercase :    Optional[int]							=							2
		_lowercase :    List[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):
												SCREAMING_SNAKE_CASE									=						0
												try:
																		SCREAMING_SNAKE_CASE									=						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      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase			):






	"""simple docstring"""





	return not any(
	    neighbour == 1 and colored_vertices[i] == color
	    for i, neighbour in enumerate(__UpperCAmelCase			)			)


def      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase			):






	"""simple docstring"""





	if index == len(__UpperCAmelCase			):
		return True

	# Recursive Step
	for i in range(__UpperCAmelCase			):
		if valid_coloring(graph[index]     ,__UpperCAmelCase     ,__UpperCAmelCase			):
			# Color current vertex
			_lowercase :    str							=							i
			# Validate coloring
			if util_color(__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase     ,index + 1			):
				return True
			# Backtrack
			_lowercase :    List[str]							=							-1
	return False







def      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase			):






	"""simple docstring"""





	_lowercase :    Any							=							[-1] * len(__UpperCAmelCase			)

	if util_color(__UpperCAmelCase     ,__UpperCAmelCase     ,__UpperCAmelCase     ,0			):
		return colored_vertices

	return []



