




"""simple docstring"""
from __future__ import annotations

import math
import random






def 				__lowerCAmelCase					(					__lowerCAmelCase			:		int  ,	__lowerCAmelCase			:		int  ,	__lowerCAmelCase			:		int						)    ->       list[int]:
 _UpperCamelCase      :	str					      =  d * e - 1
 _UpperCamelCase      :	List[Any]					      =  0
 _UpperCamelCase      :	List[Any]					      =  0
 while p == 0:
  _UpperCamelCase      :	Any					      =  random.randint(2  ,	n - 1						)
  _UpperCamelCase      :	Union[str, Any]					      =  k
  while True:
   if t % 2 == 0:
    _UpperCamelCase      :	str					      =  t // 2
    _UpperCamelCase      :	Optional[Any]					      =  (g**t) % n
    _UpperCamelCase      :	int					      =  math.gcd(x - 1  ,	__lowerCAmelCase						)
    if x > 1 and y > 1:
     _UpperCamelCase      :	Optional[Any]					      =  y
     _UpperCamelCase      :	Union[str, Any]					      =  n // y
     break  # find the correct factors
   else:
    break  # t is not divisible by 2, break and choose another g
 return sorted([p, q]						)


if __name__ == "__main__":
    import doctest

    doctest.testmod()









"""simple docstring"""
import math






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

 _UpperCamelCase      :	Tuple					      =  3

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

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

 if n > 2:
  count += 1
 return count


if __name__ == "__main__":
    _SCREAMING_SNAKE_CASE						 =     5_1_2_4_2_1_8_3
    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}')



