import math
def 					UpperCamelCase__		(							A__       )  ->   Union[str, Any]:
				if n <= 0:
								raise ValueError('Only positive integers have prime factors'       )
				snake_case__   :      Optional[int]   				= []
				while n % 2 == 0:
								pf.append(2       )
								snake_case__   :      List[Any]   				= int(n / 2       )
				for i in range(3     ,					int(math.sqrt(A__       )       ) + 1     ,					2       ):
								while n % i == 0:
												pf.append(A__       )
												snake_case__   :      Optional[int]   				= int(n / i       )
				if n > 2:
								pf.append(A__       )
				return pf
def 					UpperCamelCase__		(							A__       )  ->   int:
				if n <= 0:
								raise ValueError('Only positive numbers are accepted'       )
				snake_case__   :      str   				= 1
				snake_case__   :      Optional[Any]   				= 1
				while n % 2 == 0:
								temp += 1
								snake_case__   :      Optional[int]   				= int(n / 2       )
				div *= temp
				for i in range(3     ,					int(math.sqrt(A__       )       ) + 1     ,					2       ):
								snake_case__   :      Any   				= 1
								while n % i == 0:
												temp += 1
												snake_case__   :      Optional[int]   				= int(n / i       )
								div *= temp
				if n > 1:
								div *= 2
				return div
def 					UpperCamelCase__		(							A__       )  ->   List[str]:
				if n <= 0:
								raise ValueError('Only positive numbers are accepted'       )
				snake_case__   :      Optional[Any]   				= 1
				snake_case__   :      List[str]   				= 1
				while n % 2 == 0:
								temp += 1
								snake_case__   :      Union[str, Any]   				= int(n / 2       )
				if temp > 1:
								s *= (2**temp - 1) / (2 - 1)
				for i in range(3     ,					int(math.sqrt(A__       )       ) + 1     ,					2       ):
								snake_case__   :      Tuple   				= 1
								while n % i == 0:
												temp += 1
												snake_case__   :      Dict   				= int(n / i       )
								if temp > 1:
												s *= (i**temp - 1) / (i - 1)
				return int(A__       )
def 					UpperCamelCase__		(							A__       )  ->   Optional[Any]:
				snake_case__   :      str   				= n
				for x in set(prime_factors(A__       )       ):
								s *= (x - 1) / x
				return int(A__       )


if __name__ == "__main__":
				import doctest

				doctest.testmod()









"""simple docstring"""
from functools import lru_cache






@lru_cache
def    A__							(     UpperCamelCase			):
							if not isinstance(UpperCamelCase   ,    UpperCamelCase			):
														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    A__							(     UpperCamelCase   ,    UpperCamelCase			):
							A						=    len(UpperCamelCase			)
							# Find m such that F_m >= n where F_i is the i_th fibonacci number.
							A						=    0
							while True:
														if fibonacci(UpperCamelCase			) >= len_list:
																					A						=    i
																					break
														i += 1
							A						=    0
							while fibb_k > 0:
														A						=    min(
														    offset + fibonacci(fibb_k - 1			)   ,    len_list - 1			)  # Prevent out of range
														A						=    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()


