

import math





def 					__lowerCamelCase	( UpperCamelCase__      ):







							'''simple docstring'''






							snake_case_							=    [True] * n
							snake_case_							=    False
							snake_case_							=    False
							snake_case_							=    True

							for i in range(3   ,							int(n**0.5 + 1      )   ,							2      ):
														snake_case_							=    i * 2
														while index < n:
																					snake_case_							=    False
																					snake_case_							=    index + i

							snake_case_							=    [2]

							for i in range(3   ,							UpperCamelCase__   ,							2      ):
														if is_prime[i]:
																					primes.append(UpperCamelCase__      )

							return primes





def 					__lowerCamelCase	( UpperCamelCase__ = 999966663333      ):







							'''simple docstring'''






							snake_case_							=    math.floor(math.sqrt(UpperCamelCase__      )      ) + 100
							snake_case_							=    prime_sieve(UpperCamelCase__      )

							snake_case_							=    0
							snake_case_							=    0
							snake_case_							=    primes[prime_index]

							while (last_prime**2) <= limit:
														snake_case_							=    primes[prime_index + 1]

														snake_case_							=    last_prime**2
														snake_case_							=    next_prime**2

														# Get numbers divisible by lps(current)
														snake_case_							=    lower_bound + last_prime
														while upper_bound > current <= limit:
																					matches_sum += current
																					current += last_prime

														# Reset the upper_bound
														while (upper_bound - next_prime) > limit:
																					upper_bound -= next_prime

														# Add the numbers divisible by ups(current)
														snake_case_							=    upper_bound - next_prime
														while current > lower_bound:
																					matches_sum += current
																					current -= next_prime

														# Remove the numbers divisible by both ups and lps
														snake_case_							=    0
														while upper_bound > current <= limit:
																					if current <= lower_bound:
																												# Increment the current number
																												current += last_prime * next_prime
																												continue

																					if current > limit:
																												break

																					# Remove twice since it was added by both ups and lps
																					matches_sum -= current * 2

																					# Increment the current number
																					current += last_prime * next_prime

														# Setup for next pair
														snake_case_							=    next_prime
														prime_index += 1

							return matches_sum


if __name__ == "__main__":
					print(solution())







def 					__lowerCamelCase	( UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      ):







							'''simple docstring'''






							global f  # a global dp table for knapsack
							if f[i][j] < 0:
														if j < wt[i - 1]:
																					snake_case_							=    mf_knapsack(i - 1   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      )
														else:
																					snake_case_							=    max(
																					    mf_knapsack(i - 1   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      )   ,							mf_knapsack(i - 1   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							j - wt[i - 1]      ) + val[i - 1]   ,							)
														snake_case_							=    val
							return f[i][j]





def 					__lowerCamelCase	( UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      ):







							'''simple docstring'''






							snake_case_							=    [[0] * (w + 1) for _ in range(n + 1      )]

							for i in range(1   ,							n + 1      ):
														for w_ in range(1   ,							w + 1      ):
																					if wt[i - 1] <= w_:
																												snake_case_							=    max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]   ,							dp[i - 1][w_]      )
																					else:
																												snake_case_							=    dp[i - 1][w_]

							return dp[n][w_], dp





def 					__lowerCamelCase	( UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      ):







							'''simple docstring'''






							if not (isinstance(UpperCamelCase__   ,							(list, tuple)      ) and isinstance(UpperCamelCase__   ,							(list, tuple)      )):
														raise ValueError(
														    'Both the weights and values vectors must be either lists or tuples'      )

							snake_case_							=    len(UpperCamelCase__      )
							if num_items != len(UpperCamelCase__      ):
														snake_case_							=    (
														    'The number of weights must be the same as the number of values.\n'
														    F'''But got {num_items} weights and {len(UpperCamelCase__      )} values'''
														)
														raise ValueError(UpperCamelCase__      )
							for i in range(UpperCamelCase__      ):
														if not isinstance(wt[i]   ,							UpperCamelCase__      ):
																					snake_case_							=    (
																					    'All weights must be integers but got weight of '
																					    F'''type {type(wt[i]      )} at index {i}'''
																					)
																					raise TypeError(UpperCamelCase__      )

							snake_case_				,    snake_case_							=    knapsack(UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      )
							snake_case_							=    set()
							_construct_solution(UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      )

							return optimal_val, example_optional_set





def 					__lowerCamelCase	( UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      ):







							'''simple docstring'''






							if i > 0 and j > 0:
														if dp[i - 1][j] == dp[i][j]:
																					_construct_solution(UpperCamelCase__   ,							UpperCamelCase__   ,							i - 1   ,							UpperCamelCase__   ,							UpperCamelCase__      )
														else:
																					optimal_set.add(UpperCamelCase__      )
																					_construct_solution(UpperCamelCase__   ,							UpperCamelCase__   ,							i - 1   ,							j - wt[i - 1]   ,							UpperCamelCase__      )


if __name__ == "__main__":
					_UpperCAmelCase		:   Tuple										=       [3, 2, 4, 4]
					_UpperCAmelCase		:   Optional[Any]										=       [4, 3, 2, 3]
					_UpperCAmelCase		:   List[str]										=       4
					_UpperCAmelCase		:   str										=       6
					_UpperCAmelCase		:   Tuple										=       [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
					_UpperCAmelCase				,						_UpperCAmelCase		:   List[Any]										=       knapsack(w, wt, val, n)
					print(optimal_solution)
					print(mf_knapsack(n, wt, val, w))  # switched the n and w

					# testing the dynamic programming problem with example
					# the optimal subset for the above example are items 3 and 4
					_UpperCAmelCase				,						_UpperCAmelCase		:   Any										=       knapsack_with_example_solution(w, wt, val)
					assert optimal_solution == 8
					assert optimal_subset == {3, 4}
					print("""optimal_value = """, optimal_solution)
					print("""An optimal subset corresponding to the optimal value""", optimal_subset)




