


'''simple docstring'''

import math





def lowerCamelCase    (    lowerCAmelCase       :			int	):





	"""simple docstring"""







	if n <= 0:
		raise ValueError('Only positive integers have prime factors'	)
	__magic_name__ :					int  			=  []
	while n % 2 == 0:
		pf.append(2	)
		__magic_name__ :					List[str]  			=  int(n / 2	)
	for i in range(3	,	int(math.sqrt(lowerCAmelCase	)	) + 1	,	2	):
		while n % i == 0:
			pf.append(lowerCAmelCase	)
			__magic_name__ :					Tuple  			=  int(n / i	)
	if n > 2:
		pf.append(lowerCAmelCase	)
	return pf





def lowerCamelCase    (    lowerCAmelCase       :			int	):





	"""simple docstring"""







	if n <= 0:
		raise ValueError('Only positive numbers are accepted'	)
	__magic_name__ :					List[str]  			=  1
	__magic_name__ :					Optional[Any]  			=  1
	while n % 2 == 0:
		temp += 1
		__magic_name__ :					Optional[int]  			=  int(n / 2	)
	div *= temp
	for i in range(3	,	int(math.sqrt(lowerCAmelCase	)	) + 1	,	2	):
		__magic_name__ :					Dict  			=  1
		while n % i == 0:
			temp += 1
			__magic_name__ :					List[Any]  			=  int(n / i	)
		div *= temp
	if n > 1:
		div *= 2
	return div





def lowerCamelCase    (    lowerCAmelCase       :			int	):





	"""simple docstring"""







	if n <= 0:
		raise ValueError('Only positive numbers are accepted'	)
	__magic_name__ :					Union[str, Any]  			=  1
	__magic_name__ :					Tuple  			=  1
	while n % 2 == 0:
		temp += 1
		__magic_name__ :					str  			=  int(n / 2	)
	if temp > 1:
		s *= (2**temp - 1) / (2 - 1)
	for i in range(3	,	int(math.sqrt(lowerCAmelCase	)	) + 1	,	2	):
		__magic_name__ :					Union[str, Any]  			=  1
		while n % i == 0:
			temp += 1
			__magic_name__ :					Any  			=  int(n / i	)
		if temp > 1:
			s *= (i**temp - 1) / (i - 1)
	return int(lowerCAmelCase	)





def lowerCamelCase    (    lowerCAmelCase       :			int	):





	"""simple docstring"""







	__magic_name__ :					Union[str, Any]  			=  n
	for x in set(prime_factors(lowerCAmelCase	)	):
		s *= (x - 1) / x
	return int(lowerCAmelCase	)


if __name__ == "__main__":
							import doctest

							doctest.testmod()



'''simple docstring'''

from __future__ import annotations





def lowerCamelCase    (    lowerCAmelCase       :			list[int | float]	,	lowerCAmelCase       :			int	,	lowerCAmelCase       :			int	):





	"""simple docstring"""







	if len(lowerCAmelCase	) == 0:
		raise ValueError('find_min() arg is an empty sequence'	)
	if (
	    left >= len(lowerCAmelCase	)
	    or left < -len(lowerCAmelCase	)
	    or right >= len(lowerCAmelCase	)
	    or right < -len(lowerCAmelCase	)
	):
		raise IndexError('list index out of range'	)
	if left == right:
		return nums[left]
	__magic_name__ :					Tuple  			=  (left + right) >> 1  # the middle
	__magic_name__ :					str  			=  find_min(lowerCAmelCase	,	lowerCAmelCase	,	lowerCAmelCase	)  # find min in range[left, mid]
	__magic_name__ :					Optional[Any]  			=  find_min(lowerCAmelCase	,	mid + 1	,	lowerCAmelCase	)  # find min in range[mid + 1, right]

	return left_min if left_min <= right_min else right_min


if __name__ == "__main__":
							import doctest

							doctest.testmod(verbose=True)