


def   A   (		A__ ):
     UpperCAmelCase						:				int		  =			[True for i in range(n + 1 )]
     UpperCAmelCase						:				List[Any]		  =			[i - 1 for i in range(n + 1 )]
     UpperCAmelCase						:				Tuple		  =			[]
     for i in range(2		,		n + 1 ):
          if is_prime[i]:
               primes.append(A__ )
          for j in range(0		,		len(A__ ) ):
               if i * primes[j] >= n:
                    break
               UpperCAmelCase						:				Tuple		  =			False

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

               UpperCAmelCase						:				Dict		  =			totients[i] * (primes[j] - 1)

     return totients


if __name__ == "__main__":
      import doctest

      doctest.testmod()







from collections.abc import Callable

_snake_case   = Callable[[float], float]  # type alias for a real -> real function







def   A   (		A__		,		A__		,		A__		,		):
     UpperCAmelCase						:				Any		  =			float(A__ )
     while True:
          try:
               UpperCAmelCase						:				Optional[Any]		  =			prev_guess - function(A__ ) / derivative(A__ )
          except ZeroDivisionError:
               raise ZeroDivisionError("Could not find root" ) from None
          if abs(prev_guess - next_guess ) < 1_0**-5:
               return next_guess
          UpperCAmelCase						:				List[str]		  =			next_guess







def   A   (		A__ ):
     return (x**3) - (2 * x) - 5







def   A   (		A__ ):
     return 3 * (x**2) - 2


if __name__ == "__main__":
      print(newton(f, fa, 3))



