





import math







def       lowerCAmelCase__(	lowercase       :    List[str]				) ->    Dict:
  __snake_case   :				Dict        =       0
  if n % 2 == 0:
    count += 1
    while n % 2 == 0:
      __snake_case   :				List[str]        =       int(n / 2				)
    # the n input value must be odd so that
    # we can skip one element (ie i += 2)

  __snake_case   :				Any        =       3

  while i <= int(math.sqrt(lowercase				)				):
    if n % i == 0:
      count += 1
      while n % i == 0:
        __snake_case   :				Optional[int]        =       int(n / i				)
    __snake_case   :				List[Any]        =       i + 2

  # this condition checks the prime
  # number n is greater than 2

  if n > 2:
    count += 1
  return count


if __name__ == "__main__":
     _UpperCamelCase       = 5124_2183
     print(F'''The number of distinct prime factors is/are {exact_prime_factor_count(n)}''')
     print(F'''The value of log(log(n)) is {math.log(math.log(n)):.4f}''')








from __future__ import annotations

from collections.abc import Callable
from typing import Generic, TypeVar

_UpperCamelCase       = TypeVar('''T''')
_UpperCamelCase       = TypeVar('''U''')




class 				_lowerCamelCase (    Generic[T, U]    ):



      """simple docstring"""






      def __init__( self				,						UpperCAmelCase				,						UpperCAmelCase   )       ->   Dict:






        '''simple docstring'''






        __snake_case   :				Any        =       key
        __snake_case   :				Tuple        =       val
        __snake_case   :				int        =       0
        __snake_case   :				DoubleLinkedListNode[T, U] | None        =       None
        __snake_case   :				DoubleLinkedListNode[T, U] | None        =       None







      def __repr__( self   )       ->   str:






        '''simple docstring'''






        return "Node: key: {}, val: {}, freq: {}, has next: {}, has prev: {}".format(
            self.key				,						self.val				,						self.freq				,						self.next is not None				,						self.prev is not None   )




class 				_lowerCamelCase (    Generic[T, U]    ):



      """simple docstring"""






      def __init__( self   )       ->   None:






        '''simple docstring'''






        __snake_case   :				DoubleLinkedListNode[T, U]        =       DoubleLinkedListNode(UpperCAmelCase				,						UpperCAmelCase   )
        __snake_case   :				DoubleLinkedListNode[T, U]        =       DoubleLinkedListNode(UpperCAmelCase				,						UpperCAmelCase   )
        __snake_case    ,   __snake_case   :				Any        =       self.rear, self.head







      def __repr__( self   )       ->   str:






        '''simple docstring'''






        __snake_case   :				Union[str, Any]        =       ["DoubleLinkedList"]
        __snake_case   :				List[Any]        =       self.head
        while node.next is not None:
          rep.append(str(UpperCAmelCase   )   )
          __snake_case   :				List[Any]        =       node.next
        rep.append(str(self.rear   )   )
        return ",\n    ".join(UpperCAmelCase   )







      def 						UpperCAmelCase		( self				,						UpperCAmelCase   )       ->   None:






        '''simple docstring'''






        __snake_case   :				Dict        =       self.rear.prev

        # All nodes other than self.head are guaranteed to have non-None previous
        assert previous is not None

        __snake_case   :				int        =       node
        __snake_case   :				List[str]        =       previous
        __snake_case   :				Optional[int]        =       node
        __snake_case   :				int        =       self.rear
        node.freq += 1
        self._position_node(UpperCAmelCase   )







      def 						UpperCAmelCase		( self				,						UpperCAmelCase   )       ->   None:






        '''simple docstring'''






        while node.prev is not None and node.prev.freq > node.freq:
          # swap node with previous node
          __snake_case   :				List[Any]        =       node.prev

          __snake_case   :				Dict        =       previous_node.prev
          __snake_case   :				Union[str, Any]        =       node.prev
          __snake_case   :				Union[str, Any]        =       previous_node
          __snake_case   :				int        =       node







      def 						UpperCAmelCase		( self				,						UpperCAmelCase   )       ->   DoubleLinkedListNode[T, U] | None:






        '''simple docstring'''






        if node.prev is None or node.next is None:
          return None

        __snake_case   :				Optional[int]        =       node.next
        __snake_case   :				Optional[Any]        =       node.prev
        __snake_case   :				Optional[int]        =       None
        __snake_case   :				Any        =       None
        return node




class 				_lowerCamelCase (    Generic[T, U]    ):



      """simple docstring"""






      UpperCAmelCase_						:      dict[Callable[[T], U], LFUCache[T, U]]       ={}







      def __init__( self				,						UpperCAmelCase   )       ->   Tuple:






        '''simple docstring'''






        __snake_case   :				DoubleLinkedList[T, U]        =       DoubleLinkedList()
        __snake_case   :				str        =       capacity
        __snake_case   :				str        =       0
        __snake_case   :				Optional[int]        =       0
        __snake_case   :				Any        =       0
        __snake_case   :				dict[T, DoubleLinkedListNode[T, U]]        =       {}







      def __repr__( self   )       ->   str:






        '''simple docstring'''






        return (
            F"""CacheInfo(hits={self.hits}, misses={self.miss}, """
            F"""capacity={self.capacity}, current_size={self.num_keys})"""
        )







      def __contains__( self				,						UpperCAmelCase   )       ->   bool:






        '''simple docstring'''






        return key in self.cache







      def 						UpperCAmelCase		( self				,						UpperCAmelCase   )       ->   U | None:






        '''simple docstring'''






        if key in self.cache:
          self.hits += 1
          __snake_case   :				DoubleLinkedListNode[T, U]        =       self.cache[key]
          __snake_case   :				Union[str, Any]        =       self.list.remove(self.cache[key]   )
          assert node == value_node

          # node is guaranteed not None because it is in self.cache
          assert node is not None
          self.list.add(UpperCAmelCase   )
          return node.val
        self.miss += 1
        return None







      def 						UpperCAmelCase		( self				,						UpperCAmelCase				,						UpperCAmelCase   )       ->   None:






        '''simple docstring'''






        if key not in self.cache:
          if self.num_keys >= self.capacity:
            # delete first node when over capacity
            __snake_case   :				Optional[Any]        =       self.list.head.next

            # guaranteed to have a non-None first node when num_keys > 0
            # explain to type checker via assertions
            assert first_node is not None
            assert first_node.key is not None
            assert self.list.remove(UpperCAmelCase   ) is not None
            # first_node guaranteed to be in list

            del self.cache[first_node.key]
            self.num_keys -= 1
          __snake_case   :				Union[str, Any]        =       DoubleLinkedListNode(UpperCAmelCase				,						UpperCAmelCase   )
          self.list.add(self.cache[key]   )
          self.num_keys += 1

        else:
          __snake_case   :				Union[str, Any]        =       self.list.remove(self.cache[key]   )
          assert node is not None  # node guaranteed to be in list
          __snake_case   :				Optional[Any]        =       value
          self.list.add(UpperCAmelCase   )







      @classmethod
      def 						UpperCAmelCase		( cls				,						UpperCAmelCase = 128   )       ->   Callable[[Callable[[T], U]], Callable[..., U]]:






        '''simple docstring'''






        def cache_decorator_inner(UpperCAmelCase   ) -> Callable[..., U]:
          def cache_decorator_wrapper(*UpperCAmelCase   ) -> U:
            if func not in cls.decorator_function_to_instance_map:
              __snake_case   :				str        =       LFUCache(UpperCAmelCase   )

            __snake_case   :				Dict        =       cls.decorator_function_to_instance_map[func].get(args[0]   )
            if result is None:
              __snake_case   :				Optional[int]        =       func(*UpperCAmelCase   )
              cls.decorator_function_to_instance_map[func].put(args[0]				,						UpperCAmelCase   )
            return result

          def cache_info() -> LFUCache[T, U]:
            return cls.decorator_function_to_instance_map[func]

          setattr(UpperCAmelCase				,						"cache_info"				,						UpperCAmelCase   )  # noqa: B010

          return cache_decorator_wrapper

        return cache_decorator_inner


if __name__ == "__main__":
     import doctest

     doctest.testmod()

