code: 
  
  
  '''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()
style_context: 
  
  
  '''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)