






'''simple docstring'''




from __future__ import annotations

from collections import Counter
from random import random

class   _UpperCAmelCase     :




   """simple docstring"""

   def __init__(						self					:      Union[str, Any]						):





         '''simple docstring'''





         _A     	=      {}

   def 		lowerCAmelCase					(						self					:      List[str] , __UpperCAmelCase					:      Optional[int]						):





         '''simple docstring'''





         _A     	=      {}

   def 		lowerCAmelCase					(						self					:      Optional[int] , __UpperCAmelCase					:      Tuple , __UpperCAmelCase					:      int , __UpperCAmelCase					:      int						):





         '''simple docstring'''





         if nodea not in self.connections:
               self.add_node(_UpperCamelCase						)
         if nodea not in self.connections:
               self.add_node(_UpperCamelCase						)
         _A     	=      probability

   def 		lowerCAmelCase					(						self					:      List[Any]						):





         '''simple docstring'''





         return list(self.connections						)

   def 		lowerCAmelCase					(						self					:      int , __UpperCAmelCase					:      Optional[int]						):





         '''simple docstring'''





         _A     	=      0
         _A     	=      random()

         for dest in self.connections[node]:
               current_probability += self.connections[node][dest]
               if current_probability > random_value:
                     return dest
         return ""





def    __lowercase		(     __lowercase  ,    __lowercase  ,    __lowercase			)			->		Tuple:

      '''simple docstring'''


      _A     	=      MarkovChainGraphUndirectedUnweighted()

      for nodea, nodea, probability in transitions:
            graph.add_transition_probability(__A  ,    __A  ,    __A			)

      _A     	=      Counter(graph.get_nodes()			)
      _A     	=      start

      for _ in range(__A			):
            _A     	=      graph.transition(__A			)
            visited[node] += 1

      return visited


if __name__ == "__main__":
    import doctest

    doctest.testmod()








from math import isqrt, loga







def 						lowerCamelCase__      ( __A    :int		):


       """simple docstring"""




       __snake_case       	=							[True] * max_number
       for i in range(2	,isqrt(max_number - 1		) + 1		):
              if is_prime[i]:
                     for j in range(i**2	,__A	,__A		):
                            __snake_case       	=							False

       return [i for i in range(2	,__A		) if is_prime[i]]







def 						lowerCamelCase__      ( __A    :int = 8_0_0_8_0_0	,__A    :int = 8_0_0_8_0_0		):


       """simple docstring"""




       __snake_case       	=							degree * loga(__A		)
       __snake_case       	=							int(__A		)
       __snake_case       	=							calculate_prime_numbers(__A		)

       __snake_case       	=							0
       __snake_case       	=							0
       __snake_case       	=							len(__A		) - 1
       while left < right:
              while (
                  prime_numbers[right] * loga(prime_numbers[left]		)
                  + prime_numbers[left] * loga(prime_numbers[right]		)
                  > upper_bound
              ):
                     right -= 1
              hybrid_integers_count += right - left
              left += 1

       return hybrid_integers_count


if __name__ == "__main__":
    print(F'{solution() = }')

