

from math import sqrt

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					int	)				->    bool:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (
			    number >= 0
			), "'number' must been an int and positive"

			_UpperCAmelCase    :						str							=      True

			# 0 and 1 are none primes.
			if number <= 1:
						_UpperCAmelCase    :						int							=      False

			for divisor in range(2					, int(round(sqrt(lowerCAmelCase	)	)	) + 1	):
						# if 'number' divisible by 'divisor' then sets 'status'
						# of false and break up the loop.
						if number % divisor == 0:
									_UpperCAmelCase    :						Optional[int]							=      False
									break

    # precondition
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	), "'status' must been from type bool"

			return status

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					Tuple	)				->    Optional[int]:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (n > 2), "'N' must been an int and > 2"

			# beginList: contains all natural numbers from 2 up to N
			_UpperCAmelCase    :						Optional[int]							=      list(range(2					, n + 1	)	)

			_UpperCAmelCase    :						Any							=      []  # this list will be returns.

			# actual sieve of erathostenes
			for i in range(len(lowerCAmelCase	)	):
						for j in range(i + 1					, len(lowerCAmelCase	)	):
									if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
												_UpperCAmelCase    :						Optional[Any]							=      0

    # filters actual prime numbers.
			_UpperCAmelCase    :						Tuple							=      [x for x in begin_list if x != 0]

			# precondition
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	), "'ans' must been from type list"

			return ans

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					List[Any]	)				->    Optional[int]:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (n > 2), "'N' must been an int and > 2"

			_UpperCAmelCase    :						Any							=      []

			# iterates over all numbers between 2 up to N+1
			# if a number is prime then appends to list 'ans'
			for number in range(2					, n + 1	):
						if is_prime(lowerCAmelCase	):
									ans.append(lowerCAmelCase	)

    # precondition
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	), "'ans' must been from type list"

			return ans

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					int	)				->    int:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and number >= 0, "'number' must been an int and >= 0"

			_UpperCAmelCase    :						Union[str, Any]							=      []  # this list will be returns of the function.

			# potential prime number factors.

			_UpperCAmelCase    :						str							=      2

			_UpperCAmelCase    :						Optional[int]							=      number

			if number == 0 or number == 1:
						ans.append(lowerCAmelCase	)

			# if 'number' not prime then builds the prime factorization of 'number'
			elif not is_prime(lowerCAmelCase	):
						while quotient != 1:
									if is_prime(lowerCAmelCase	) and (quotient % factor == 0):
												ans.append(lowerCAmelCase	)
												quotient /= factor
									else:
												factor += 1

			else:
						ans.append(lowerCAmelCase	)

			# precondition
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	), "'ans' must been from type list"

			return ans

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					Any	)				->    str:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (
			    number >= 0
			), "'number' bust been an int and >= 0"

			_UpperCAmelCase    :						str							=      0

			# prime factorization of 'number'
			_UpperCAmelCase    :						Any							=      prime_factorization(lowerCAmelCase	)

			_UpperCAmelCase    :						Any							=      max(lowerCAmelCase	)

			# precondition
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	), "'ans' must been from type int"

			return ans

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					Optional[Any]	)				->    List[str]:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (
			    number >= 0
			), "'number' bust been an int and >= 0"

			_UpperCAmelCase    :						Optional[Any]							=      0

			# prime factorization of 'number'
			_UpperCAmelCase    :						Dict							=      prime_factorization(lowerCAmelCase	)

			_UpperCAmelCase    :						Union[str, Any]							=      min(lowerCAmelCase	)

			# precondition
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	), "'ans' must been from type int"

			return ans

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					List[str]	)				->    Any:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	), "'number' must been an int"
			assert isinstance(number % 2 == 0					, lowerCAmelCase	), "compare bust been from type bool"

			return number % 2 == 0

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					Tuple	)				->    List[str]:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	), "'number' must been an int"
			assert isinstance(number % 2 != 0					, lowerCAmelCase	), "compare bust been from type bool"

			return number % 2 != 0

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					Dict	)				->    Optional[Any]:
			assert (
			    isinstance(lowerCAmelCase					, lowerCAmelCase	) and (number > 2) and is_even(lowerCAmelCase	)
			), "'number' must been an int, even and > 2"

			_UpperCAmelCase    :						int							=      []  # this list will returned

			# creates a list of prime numbers between 2 up to 'number'
			_UpperCAmelCase    :						Union[str, Any]							=      get_prime_numbers(lowerCAmelCase	)
			_UpperCAmelCase    :						str							=      len(lowerCAmelCase	)

			# run variable for while-loops.
			_UpperCAmelCase    :						List[str]							=      0
			_UpperCAmelCase    :						Dict							=      None

			# exit variable. for break up the loops
			_UpperCAmelCase    :						List[Any]							=      True

			while i < len_pn and loop:
						_UpperCAmelCase    :						Optional[Any]							=      i + 1

						while j < len_pn and loop:
									if prime_numbers[i] + prime_numbers[j] == number:
												_UpperCAmelCase    :						Optional[Any]							=      False
												ans.append(prime_numbers[i]	)
												ans.append(prime_numbers[j]	)

									j += 1

						i += 1

			# precondition
			assert (
			    isinstance(lowerCAmelCase					, lowerCAmelCase	)
			    and (len(lowerCAmelCase	) == 2)
			    and (ans[0] + ans[1] == number)
			    and is_prime(ans[0]	)
			    and is_prime(ans[1]	)
			), "'ans' must contains two primes. And sum of elements must been eq 'number'"

			return ans

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					int					, lowerCAmelCase:					Optional[Any]	)				->    Optional[Any]:
			assert (
			    isinstance(lowerCAmelCase					, lowerCAmelCase	)
			    and isinstance(lowerCAmelCase					, lowerCAmelCase	)
			    and (numbera >= 0)
			    and (numbera >= 0)
			), "'number1' and 'number2' must been positive integer."

			_UpperCAmelCase    :						List[str]							=      0

			while numbera != 0:
						_UpperCAmelCase    :						Optional[Any]							=      numbera % numbera
						_UpperCAmelCase    :						List[Any]							=      numbera
						_UpperCAmelCase    :						Dict							=      rest

			# precondition
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (
			    numbera >= 0
			), "'number' must been from type int and positive"

			return numbera

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					List[Any]					, lowerCAmelCase:					Any	)				->    Dict:
			assert (
			    isinstance(lowerCAmelCase					, lowerCAmelCase	)
			    and isinstance(lowerCAmelCase					, lowerCAmelCase	)
			    and (numbera >= 1)
			    and (numbera >= 1)
			), "'number1' and 'number2' must been positive integer."

			_UpperCAmelCase    :						Any							=      1  # actual answer that will be return.

			# for kgV (x,1)
			if numbera > 1 and numbera > 1:
						# builds the prime factorization of 'number1' and 'number2'
						_UpperCAmelCase    :						List[Any]							=      prime_factorization(lowerCAmelCase	)
						_UpperCAmelCase    :						Union[str, Any]							=      prime_factorization(lowerCAmelCase	)

			elif numbera == 1 or numbera == 1:
						_UpperCAmelCase    :						Optional[Any]							=      []
						_UpperCAmelCase    :						Optional[Any]							=      []
						_UpperCAmelCase    :						Dict							=      max(lowerCAmelCase					, lowerCAmelCase	)

			_UpperCAmelCase    :						int							=      0
			_UpperCAmelCase    :						int							=      0

			_UpperCAmelCase    :						Union[str, Any]							=      []  # captured numbers int both 'primeFac1' and 'primeFac2'

			# iterates through primeFac1
			for n in prime_fac_a:
						if n not in done:
									if n in prime_fac_a:
												_UpperCAmelCase    :						Union[str, Any]							=      prime_fac_a.count(lowerCAmelCase	)
												_UpperCAmelCase    :						int							=      prime_fac_a.count(lowerCAmelCase	)

												for _ in range(max(lowerCAmelCase					, lowerCAmelCase	)	):
															ans *= n

									else:
												_UpperCAmelCase    :						Dict							=      prime_fac_a.count(lowerCAmelCase	)

												for _ in range(lowerCAmelCase	):
															ans *= n

									done.append(lowerCAmelCase	)

    # iterates through primeFac2
			for n in prime_fac_a:
						if n not in done:
									_UpperCAmelCase    :						Any							=      prime_fac_a.count(lowerCAmelCase	)

									for _ in range(lowerCAmelCase	):
												ans *= n

									done.append(lowerCAmelCase	)

    # precondition
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (
			    ans >= 0
			), "'ans' must been from type int and positive"

			return ans

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					int	)				->    Optional[Any]:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (n >= 0), "'number' must been a positive int"

			_UpperCAmelCase    :						Tuple							=      0
			_UpperCAmelCase    :						Union[str, Any]							=      2  # this variable holds the answer

			while index < n:
						index += 1

						ans += 1  # counts to the next number

						# if ans not prime then
						# runs to the next prime number.
						while not is_prime(lowerCAmelCase	):
									ans += 1

    # precondition
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and is_prime(
			    lowerCAmelCase	), "'ans' must been a prime number and from type int"

			return ans

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					Union[str, Any]					, lowerCAmelCase:					Optional[int]	)				->    str:
			assert (
			    is_prime(lowerCAmelCase	) and is_prime(lowerCAmelCase	) and (p_number_a < p_number_a)
			), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

			_UpperCAmelCase    :						List[str]							=      p_number_a + 1  # jump to the next number

			_UpperCAmelCase    :						List[str]							=      []  # this list will be returns.

			# if number is not prime then
			# fetch the next prime number.
			while not is_prime(lowerCAmelCase	):
						number += 1

			while number < p_number_a:
						ans.append(lowerCAmelCase	)

						number += 1

						# fetch the next prime number.
						while not is_prime(lowerCAmelCase	):
									number += 1

    # precondition
			assert (
			    isinstance(lowerCAmelCase					, lowerCAmelCase	)
			    and ans[0] != p_number_a
			    and ans[len(lowerCAmelCase	) - 1] != p_number_a
			), "'ans' must been a list without the arguments"

			# 'ans' contains not 'pNumber1' and 'pNumber2' !
			return ans

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					Any	)				->    List[Any]:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (n >= 1), "'n' must been int and >= 1"

			_UpperCAmelCase    :						List[str]							=      []  # will be returned.

			for divisor in range(1					, n + 1	):
						if n % divisor == 0:
									ans.append(lowerCAmelCase	)

    # precondition
			assert ans[0] == 1 and ans[len(lowerCAmelCase	) - 1] == n, "Error in function getDivisiors(...)"

			return ans

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					Optional[int]	)				->    List[Any]:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (
			    number > 1
			), "'number' must been an int and >= 1"

			_UpperCAmelCase    :						Any							=      get_divisors(lowerCAmelCase	)

			# precondition
			assert (
			    isinstance(lowerCAmelCase					, lowerCAmelCase	)
			    and (divisors[0] == 1)
			    and (divisors[len(lowerCAmelCase	) - 1] == number)
			), "Error in help-function getDivisiors(...)"

			# summed all divisors up to 'number' (exclusive), hence [:-1]
			return sum(divisors[:-1]	) == number

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					Optional[Any]					, lowerCAmelCase:					Optional[int]	)				->    str:
			assert (
			    isinstance(lowerCAmelCase					, lowerCAmelCase	)
			    and isinstance(lowerCAmelCase					, lowerCAmelCase	)
			    and (denominator != 0)
			), "The arguments must been from type int and 'denominator' != 0"

			# build the greatest common divisor of numerator and denominator.
			_UpperCAmelCase    :						Tuple							=      gcd(abs(lowerCAmelCase	)					, abs(lowerCAmelCase	)	)

			# precondition
			assert (
			    isinstance(lowerCAmelCase					, lowerCAmelCase	)
			    and (numerator % gcd_of_fraction == 0)
			    and (denominator % gcd_of_fraction == 0)
			), "Error in function gcd(...,...)"

			return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)

def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					Any	)				->    List[str]:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (n >= 0), "'n' must been a int and >= 0"

			_UpperCAmelCase    :						int							=      1  # this will be return.

			for factor in range(1					, n + 1	):
						ans *= factor

			return ans





def 							__SCREAMING_SNAKE_CASE							(			lowerCAmelCase:					List[str]	)				->    List[Any]:
			assert isinstance(lowerCAmelCase					, lowerCAmelCase	) and (n >= 0), "'n' must been an int and >= 0"

			_UpperCAmelCase    :						List[Any]							=      0
			_UpperCAmelCase    :						Union[str, Any]							=      1
			_UpperCAmelCase    :						Dict							=      1  # this will be return

			for _ in range(n - 1	):
						_UpperCAmelCase    :						Optional[int]							=      ans
						ans += fiba
						_UpperCAmelCase    :						Optional[Any]							=      tmp

			return ans





from __future__ import annotations

class 			a					:


						def __init__(    self  ,   A_		):







										'''simple docstring'''


										_UpperCAmelCase    :						Union[str, Any]							=      order

										# a_{0} ... a_{k}
										_UpperCAmelCase    :						Tuple							=      [1.0] + [0.0] * order
										# b_{0} ... b_{k}
										_UpperCAmelCase    :						int							=      [1.0] + [0.0] * order

										# x[n-1] ... x[n-k]
										_UpperCAmelCase    :						Optional[Any]							=      [0.0] * self.order
										# y[n-1] ... y[n-k]
										_UpperCAmelCase    :						Dict							=      [0.0] * self.order


						def     _UpperCAmelCase			(    self  ,   A_  ,   A_		):







										'''simple docstring'''


										if len(A_		) < self.order:
														_UpperCAmelCase    :						List[str]							=      [1.0, *a_coeffs]

										if len(A_		) != self.order + 1:
														_UpperCAmelCase    :						List[Any]							=      (
														    f'Expected a_coeffs to have {self.order + 1} elements '
														    f'for {self.order}-order filter, got {len(A_		)}'
														)
														raise ValueError(A_		)

										if len(A_		) != self.order + 1:
														_UpperCAmelCase    :						int							=      (
														    f'Expected b_coeffs to have {self.order + 1} elements '
														    f'for {self.order}-order filter, got {len(A_		)}'
														)
														raise ValueError(A_		)

										_UpperCAmelCase    :						Optional[Any]							=      a_coeffs
										_UpperCAmelCase    :						Union[str, Any]							=      b_coeffs





						def     _UpperCAmelCase			(    self  ,   A_		):







										'''simple docstring'''


										_UpperCAmelCase    :						Tuple							=      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]
														)

										_UpperCAmelCase    :						Dict							=      (result + self.b_coeffs[0] * sample) / self.a_coeffs[0]

										_UpperCAmelCase    :						Optional[Any]							=      self.input_history[:-1]
										_UpperCAmelCase    :						Optional[int]							=      self.output_history[:-1]

										_UpperCAmelCase    :						Optional[Any]							=      sample
										_UpperCAmelCase    :						str							=      result

										return result


