






"""simple docstring"""




from __future__ import annotations

import re






def 							lowerCAmelCase			(   UpperCamelCase_:				list[str]       )							->				list[str]:







  '''simple docstring'''







  def alphanum_key(UpperCamelCase_:				Optional[int]       ):
    return [int(UpperCamelCase_       ) if s.isdigit() else s.lower() for s in re.split("([0-9]+)"  ,			UpperCamelCase_       )]

  return sorted(UpperCamelCase_  ,			key=UpperCamelCase_       )


if __name__ == "__main__":
     import doctest

     doctest.testmod()










"""simple docstring"""




import math






def 							lowerCAmelCase			(   UpperCamelCase_:				int       )							->				list:







  '''simple docstring'''







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






def 							lowerCAmelCase			(   UpperCamelCase_:				int       )							->				int:







  '''simple docstring'''







  if n <= 0:
    raise ValueError("Only positive numbers are accepted"       )
  _a					      =						1
  _a					      =						1
  while n % 2 == 0:
    temp += 1
    _a					      =						int(n / 2       )
  div *= temp
  for i in range(3  ,			int(math.sqrt(UpperCamelCase_       )       ) + 1  ,			2       ):
    _a					      =						1
    while n % i == 0:
      temp += 1
      _a					      =						int(n / i       )
    div *= temp
  if n > 1:
    div *= 2
  return div






def 							lowerCAmelCase			(   UpperCamelCase_:				int       )							->				int:







  '''simple docstring'''







  if n <= 0:
    raise ValueError("Only positive numbers are accepted"       )
  _a					      =						1
  _a					      =						1
  while n % 2 == 0:
    temp += 1
    _a					      =						int(n / 2       )
  if temp > 1:
    s *= (2**temp - 1) / (2 - 1)
  for i in range(3  ,			int(math.sqrt(UpperCamelCase_       )       ) + 1  ,			2       ):
    _a					      =						1
    while n % i == 0:
      temp += 1
      _a					      =						int(n / i       )
    if temp > 1:
      s *= (i**temp - 1) / (i - 1)
  return int(UpperCamelCase_       )






def 							lowerCAmelCase			(   UpperCamelCase_:				int       )							->				int:







  '''simple docstring'''







  _a					      =						n
  for x in set(prime_factors(UpperCamelCase_       )       ):
    s *= (x - 1) / x
  return int(UpperCamelCase_       )


if __name__ == "__main__":
     import doctest

     doctest.testmod()


