






from functools import lru_cache
@lru_cache
def 							UpperCAmelCase   ( _lowercase   ):


    '''simple docstring'''



    if not isinstance(_lowercase,   _lowercase   ):
        raise TypeError('k must be an integer.'   )
    if k < 0:
        raise ValueError('k integer must be greater or equal to zero.'   )
    if k == 0:
        return 0
    elif k == 1:
        return 1
    else:
        return fibonacci(k - 1   ) + fibonacci(k - 2   )
def 							UpperCAmelCase   ( _lowercase,   _lowercase   ):


    '''simple docstring'''



    _lowercase     			=						len(_lowercase   )
    # Find m such that F_m >= n where F_i is the i_th fibonacci number.
    _lowercase     			=						0
    while True:
        if fibonacci(_lowercase   ) >= len_list:
            _lowercase     			=						i
            break
        i += 1
    _lowercase     			=						0
    while fibb_k > 0:
        _lowercase     			=						min(
            offset + fibonacci(fibb_k - 1   ),   len_list - 1   )  # Prevent out of range
        _lowercase     			=						arr[index_k]
        if item_k_1 == val:
            return index_k
        elif val < item_k_1:
            fibb_k -= 1
        elif val > item_k_1:
            offset += fibonacci(fibb_k - 1   )
            fibb_k -= 2
    else:
        return -1


if __name__ == "__main__":
   import doctest

   doctest.testmod()




def 			__lowerCAmelCase       (SCREAMING_SNAKE_CASE   )->    list:




      """simple docstring"""

      snake_case_			= [True for i in range(n + 1   )]
      snake_case_			= [i - 1 for i in range(n + 1   )]
      snake_case_			= []
      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
                  snake_case_			= False

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

                  snake_case_			= totients[i] * (primes[j] - 1)

      return totients


if __name__ == "__main__":
   import doctest

   doctest.testmod()