'''simple docstring'''
from functools import lru_cache







@lru_cache
def 							__UpperCamelCase   (				__UpperCAmelCase     :				List[Any] )       ->  int:





		if not isinstance(__a   ,      __a ):
				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   (				__UpperCAmelCase     :				Dict   ,      __UpperCAmelCase     :				Optional[int] )       ->  int:
		lowercase							:       Any											=      len(__a )
		# Find m such that F_m >= n where F_i is the i_th fibonacci number.
		lowercase							:       Dict											=      0
		while True:
				if fibonacci(__a ) >= len_list:
						lowercase							:       List[Any]											=      i
						break
				i += 1
		lowercase							:       Tuple											=      0
		while fibb_k > 0:
				lowercase							:       Optional[int]											=      min(
				    offset + fibonacci(fibb_k - 1 )   ,      len_list - 1 )  # Prevent out of range
				lowercase							:       Any											=      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()
'''simple docstring'''
def 							__UpperCamelCase   (				__UpperCAmelCase     :				int )       ->  list:
  lowercase							:       Union[str, Any]											=      [True for i in range(n + 1 )]
  lowercase							:       Dict											=      [i - 1 for i in range(n + 1 )]
  lowercase							:       Union[str, Any]											=      []
  for i in range(2   ,      n + 1 ):
    if is_prime[i]:
      primes.append(__UpperCAmelCase )
    for j in range(0   ,      len(__UpperCAmelCase ) ):
      if i * primes[j] >= n:
        break
      lowercase							:       List[str]											=      False

      if i % primes[j] == 0:
        lowercase							:       Optional[Any]											=      totients[i] * primes[j]
        break

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

  return totients


if __name__ == "__main__":
 import doctest

 doctest.testmod()