
"""simple docstring"""





from math import pow



def 						_lowerCAmelCase				(				UpperCamelCase_  ,				UpperCamelCase_  ,				UpperCamelCase_  ,				UpperCamelCase_  ,				UpperCamelCase_  ,				):
		if current_sum == needed_sum:
				# If the sum of the powers is equal to needed_sum, then we have a solution.
				solutions_count += 1
				return current_sum, solutions_count

		__SCREAMING_SNAKE_CASE            = int(pow(UpperCamelCase_  ,				UpperCamelCase_				)				)
		if current_sum + i_to_n <= needed_sum:
				# If the sum of the powers is less than needed_sum, then continue adding powers.
				current_sum += i_to_n
				__SCREAMING_SNAKE_CASE       ,__SCREAMING_SNAKE_CASE            = backtrack(
				    UpperCamelCase_  ,				UpperCamelCase_  ,				current_number + 1  ,				UpperCamelCase_  ,				UpperCamelCase_				)
				current_sum -= i_to_n
		if i_to_n < needed_sum:
				# If the power of i is less than needed_sum, then try with the next power.
				__SCREAMING_SNAKE_CASE       ,__SCREAMING_SNAKE_CASE            = backtrack(
				    UpperCamelCase_  ,				UpperCamelCase_  ,				current_number + 1  ,				UpperCamelCase_  ,				UpperCamelCase_				)
		return current_sum, solutions_count



def 						_lowerCAmelCase				(				UpperCamelCase_  ,				UpperCamelCase_				):
		if not (1 <= needed_sum <= 1000 and 2 <= power <= 10):
				raise ValueError(
				    """Invalid input\n"""
				    """needed_sum must be between 1 and 1000, power between 2 and 10."""				)

		return backtrack(UpperCamelCase_  ,				UpperCamelCase_  ,				1  ,				0  ,				0				)[1]  # Return the solutions_count


if __name__ == "__main__":
							import doctest

							doctest.testmod()








"""simple docstring"""







from __future__ import annotations

from functools import lru_cache
from math import ceil

lowerCAmelCase__      				=   100

lowerCAmelCase__      				=   set(range(3, NUM_PRIMES, 2))
primes.add(2)
lowerCAmelCase__      				=   42

for prime in range(3, ceil(NUM_PRIMES**0.5), 2):
     if prime not in primes:
          continue
     primes.difference_update(set(range(prime * prime, NUM_PRIMES, prime)))
@lru_cache(maxsize=1_0_0				)
def 					a__			(				SCREAMING_SNAKE_CASE    :    int				):


  '''simple docstring'''







  if number_to_partition < 0:
    return set()
  elif number_to_partition == 0:
    return {1}

  lowerCAmelCase     :     set[int]								=							set()
  lowerCAmelCase     :     int
  lowerCAmelCase     :     int

  for prime in primes:
    if prime > number_to_partition:
      continue
    for sub in partition(number_to_partition - prime				):
      ret.add(sub * prime				)

  return ret
def 					a__			(				SCREAMING_SNAKE_CASE    :    int = 5_0_0_0				):


  '''simple docstring'''







  for number_to_partition in range(1				,      SCREAMING_SNAKE_CASE				):
    if len(partition(SCREAMING_SNAKE_CASE				)				) > number_unique_partitions:
      return number_to_partition
  return None


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





