def 	__lowerCamelCase							(UpperCAmelCase__     :							int  ):
					assert isinstance(UpperCAmelCase__      , UpperCAmelCase__  ), F"The input value of [n={number}] is not an integer"

					if number == 1:
										return 2
					elif number < 1:
										SCREAMING_SNAKE_CASE    			=       F"The input value of [n={number}] has to be > 0"
										raise ValueError(UpperCAmelCase__  )
					else:
										SCREAMING_SNAKE_CASE    			=       sylvester(number - 1  )
										SCREAMING_SNAKE_CASE    			=       num - 1
										SCREAMING_SNAKE_CASE    			=       num
										return lower * upper + 1


if __name__ == "__main__":
							print(f"""The 8th number in Sylvester's sequence: {sylvester(8)}""")





from __future__ import annotations

import math





def 	__lowerCamelCase							(UpperCAmelCase__     :							int  ):

					if 1 < number < 4:
										# 2 and 3 are primes
										return True
					elif number < 2 or number % 2 == 0 or number % 3 == 0:
										# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
										return False

					# All primes number are in format of 6k +/- 1
					for i in range(5      , int(math.sqrt(UpperCAmelCase__  ) + 1  )      , 6  ):
										if number % i == 0 or number % (i + 2) == 0:
															return False
					return True


_lowerCamelCase					:   Tuple							=		[num for num in range(3, 10_00_01, 2) if not is_prime(num)]





def 	__lowerCamelCase							(UpperCAmelCase__     :							int  ):
					if not isinstance(UpperCAmelCase__      , UpperCAmelCase__  ):
										raise ValueError("n must be an integer"  )
					if n <= 0:
										raise ValueError("n must be >= 0"  )

					SCREAMING_SNAKE_CASE    			=       []
					for num in range(len(UpperCAmelCase__  )  ):
										SCREAMING_SNAKE_CASE    			=       0
										while 2 * i * i <= odd_composites[num]:
															SCREAMING_SNAKE_CASE    			=       odd_composites[num] - 2 * i * i
															if is_prime(UpperCAmelCase__  ):
																				break
															i += 1
										else:
															list_nums.append(odd_composites[num]  )
															if len(UpperCAmelCase__  ) == n:
																				return list_nums

					return []





def 	__lowerCamelCase							():
					return compute_nums(1  )[0]


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




