


def 			_a      (    __UpperCamelCase			):
     a_     :  Tuple				      =     [True for i in range(n + 1			)]
     a_     :  Tuple				      =     [i - 1 for i in range(n + 1			)]
     a_     :  Optional[Any]				      =     []
     for i in range(2	,	n + 1			):
          if is_prime[i]:
               primes.append(SCREAMING_SNAKE_CASE_			)
          for j in range(0	,	len(SCREAMING_SNAKE_CASE_			)			):
               if i * primes[j] >= n:
                    break
               a_     :  Any				      =     False

               if i % primes[j] == 0:
                    a_     :  int				      =     totients[i] * primes[j]
                    break

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

     return totients


if __name__ == "__main__":
       import doctest

       doctest.testmod()







import sys


def   a						(				SCREAMING_SNAKE_CASE_   :		int				,      SCREAMING_SNAKE_CASE_   :		list[int]							):






       """simple docstring"""





       if not numbers:
              return 0
       if target == 0 and target in numbers:
              return 0
       if not isinstance(SCREAMING_SNAKE_CASE_				,      (list, tuple)							) or not all(
           isinstance(SCREAMING_SNAKE_CASE_				,      SCREAMING_SNAKE_CASE_							) for number in numbers							):
              raise ValueError('''numbers must be an iterable of integers'''							)

       UpperCamelCase     :						Dict 						=		0
       UpperCamelCase     :						Tuple 						=		sys.maxsize

       while right < len(SCREAMING_SNAKE_CASE_							):
              curr_sum += numbers[right]
              while curr_sum >= target and left <= right:
                     UpperCamelCase     :						int 						=		min(SCREAMING_SNAKE_CASE_				,      right - left + 1							)
                     curr_sum -= numbers[left]
                     left += 1
              right += 1

       return 0 if min_len == sys.maxsize else min_len

