






"""simple docstring"""



def   SCREAMING_SNAKE_CASE__					(     snake_case					:   list							)->		bool:






     '''simple docstring'''



     if not isinstance(snake_case ,				snake_case							):
          raise ValueError("Input series is not valid, valid series - [2, 4, 8]"							)
     if len(snake_case							) == 0:
          raise ValueError("Input list must be a non empty list"							)
     if len(snake_case							) == 1:
          return True
     try:
          UpperCAmelCase__	:						Union[str, Any]       =       series[1] / series[0]
          for index in range(len(snake_case							) - 1							):
               if series[index + 1] / series[index] != common_ratio:
                    return False
     except ZeroDivisionError:
          return False
     return True


def   SCREAMING_SNAKE_CASE__					(     snake_case					:   list							)->		float:






     '''simple docstring'''



     if not isinstance(snake_case ,				snake_case							):
          raise ValueError("Input series is not valid, valid series - [2, 4, 8]"							)
     if len(snake_case							) == 0:
          raise ValueError("Input list must be a non empty list"							)
     UpperCAmelCase__	:						Union[str, Any]       =       1
     for value in series:
          answer *= value
     return pow(snake_case ,				1 / len(snake_case							)							)


if __name__ == "__main__":
 import doctest

 doctest.testmod()










"""simple docstring"""



def   SCREAMING_SNAKE_CASE__					(     snake_case					:   int							)->		list:






     '''simple docstring'''



     UpperCAmelCase__	:						List[str]       =       [True for i in range(n + 1							)]
     UpperCAmelCase__	:						List[str]       =       [i - 1 for i in range(n + 1							)]
     UpperCAmelCase__	:						int       =       []
     for i in range(2 ,				n + 1							):
          if is_prime[i]:
               primes.append(snake_case							)
          for j in range(0 ,				len(snake_case							)							):
               if i * primes[j] >= n:
                    break
               UpperCAmelCase__	:						Optional[Any]       =       False

               if i % primes[j] == 0:
                    UpperCAmelCase__	:						str       =       totients[i] * primes[j]
                    break

               UpperCAmelCase__	:						Union[str, Any]       =       totients[i] * (primes[j] - 1)

     return totients


if __name__ == "__main__":
 import doctest

 doctest.testmod()


