from __future__ import annotations
class   _lowerCAmelCase		:







       def __init__(				self     :    Dict	,					__snake_case     :    int		):
          lowerCamelCase						:Tuple						  =     order

          # a_{0} ... a_{k}
          lowerCamelCase						:Dict						  =     [1.0] + [0.0] * order
          # b_{0} ... b_{k}
          lowerCamelCase						:Optional[int]						  =     [1.0] + [0.0] * order

          # x[n-1] ... x[n-k]
          lowerCamelCase						:Union[str, Any]						  =     [0.0] * self.order
          # y[n-1] ... y[n-k]
          lowerCamelCase						:Any						  =     [0.0] * self.order







       def 				snake_case  (				self     :    List[str]	,					__snake_case     :    list[float]	,					__snake_case     :    list[float]		):
          if len(__snake_case		) < self.order:
             lowerCamelCase						:int						  =     [1.0, *a_coeffs]

          if len(__snake_case		) != self.order + 1:
             lowerCamelCase						:str						  =     (
                 F"Expected a_coeffs to have {self.order + 1} elements "
                 F"for {self.order}-order filter, got {len(__snake_case		)}"
             )
             raise ValueError(__snake_case		)

          if len(__snake_case		) != self.order + 1:
             lowerCamelCase						:str						  =     (
                 F"Expected b_coeffs to have {self.order + 1} elements "
                 F"for {self.order}-order filter, got {len(__snake_case		)}"
             )
             raise ValueError(__snake_case		)

          lowerCamelCase						:Optional[int]						  =     a_coeffs
          lowerCamelCase						:Tuple						  =     b_coeffs







       def 				snake_case  (				self     :    Tuple	,					__snake_case     :    float		):
          lowerCamelCase						:int						  =     0.0

          # Start at index 1 and do index 0 at the end.
          for i in range(1	,					self.order + 1		):
             result += (
                 self.b_coeffs[i] * self.input_history[i - 1]
                 - self.a_coeffs[i] * self.output_history[i - 1]
             )

          lowerCamelCase						:Union[str, Any]						  =     (result + self.b_coeffs[0] * sample) / self.a_coeffs[0]

          lowerCamelCase						:Optional[Any]						  =     self.input_history[:-1]
          lowerCamelCase						:Union[str, Any]						  =     self.output_history[:-1]

          lowerCamelCase						:int						  =     sample
          lowerCamelCase						:int						  =     result

          return result




import math





def  _lowerCamelCase					(   a_				: 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(a_) + 1)						,			6):
						if number % i == 0 or number % (i + 2) == 0:
									return False
			return True





def  _lowerCamelCase					(   a_				: float = 0.1):
			lowerCamelCase						:Dict						  =     3
			lowerCamelCase						:List[Any]						  =     3

			while primes / (2 * j - 1) >= ratio:
						for i in range(j * j + j + 1						,			(j + 2) * (j + 2)						,			j + 1):
									primes += is_prime(a_)
						j += 2
			return j


if __name__ == "__main__":
			import doctest

			doctest.testmod()



