
from math import sqrt




def    __UpperCamelCase     (				lowerCAmelCase__							:				int  ):
 assert isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and (
     number >= 0
 ), "'number' must been an int and positive"

 __a :  str						=  True

 # 0 and 1 are none primes.
 if number <= 1:
  __a :  str						=  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:
   __a :  List[Any]						=  False
   break

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

 return status




def    __UpperCamelCase     (				lowerCAmelCase__							:				Tuple  ):
 assert isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and (n > 2), "'N' must been an int and > 2"

 # beginList: contains all natural numbers from 2 up to N
 __a :  Dict						=  list(range(2  ,   n + 1  )  )

 __a :  Union[str, 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):
    __a :  Any						=  0

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

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

 return ans




def    __UpperCamelCase     (				lowerCAmelCase__							:				Dict  ):
 assert isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and (n > 2), "'N' must been an int and > 2"

 __a :  str						=  []

 # 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    __UpperCamelCase     (				lowerCAmelCase__							:				str  ):
 assert isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and number >= 0, "'number' must been an int and >= 0"

 __a :  Any						=  []  # this list will be returns of the function.

 # potential prime number factors.

 __a :  List[str]						=  2

 __a :  Any						=  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    __UpperCamelCase     (				lowerCAmelCase__							:				Optional[Any]  ):
 assert isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and (
     number >= 0
 ), "'number' bust been an int and >= 0"

 __a :  Any						=  0

 # prime factorization of 'number'
 __a :  Optional[int]						=  prime_factorization(lowerCAmelCase__  )

 __a :  Any						=  max(lowerCAmelCase__  )

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

 return ans




def    __UpperCamelCase     (				lowerCAmelCase__							:				List[Any]  ):
 assert isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and (
     number >= 0
 ), "'number' bust been an int and >= 0"

 __a :  List[Any]						=  0

 # prime factorization of 'number'
 __a :  Tuple						=  prime_factorization(lowerCAmelCase__  )

 __a :  Tuple						=  min(lowerCAmelCase__  )

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

 return ans




def    __UpperCamelCase     (				lowerCAmelCase__							:				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    __UpperCamelCase     (				lowerCAmelCase__							:				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    __UpperCamelCase     (				lowerCAmelCase__							:				Union[str, Any]  ):
 assert (
     isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and (number > 2) and is_even(lowerCAmelCase__  )
 ), "'number' must been an int, even and > 2"

 __a :  int						=  []  # this list will returned

 # creates a list of prime numbers between 2 up to 'number'
 __a :  List[Any]						=  get_prime_numbers(lowerCAmelCase__  )
 __a :  Tuple						=  len(lowerCAmelCase__  )

 # run variable for while-loops.
 __a :  Optional[Any]						=  0
 __a :  Dict						=  None

 # exit variable. for break up the loops
 __a :  Any						=  True

 while i < len_pn and loop:
  __a :  Any						=  i + 1

  while j < len_pn and loop:
   if prime_numbers[i] + prime_numbers[j] == number:
    __a :  int						=  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    __UpperCamelCase     (				lowerCAmelCase__							:				Any  ,   lowerCAmelCase__							:				int  ):
 assert (
     isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  )
     and isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  )
     and (numbera >= 0)
     and (numbera >= 0)
 ), "'number1' and 'number2' must been positive integer."

 __a :  Dict						=  0

 while numbera != 0:
  __a :  Tuple						=  numbera % numbera
  __a :  Any						=  numbera
  __a :  str						=  rest

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

 return numbera




def    __UpperCamelCase     (				lowerCAmelCase__							:				Optional[int]  ,   lowerCAmelCase__							:				List[Any]  ):
 assert (
     isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  )
     and isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  )
     and (numbera >= 1)
     and (numbera >= 1)
 ), "'number1' and 'number2' must been positive integer."

 __a :  str						=  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'
  __a :  Any						=  prime_factorization(lowerCAmelCase__  )
  __a :  Optional[int]						=  prime_factorization(lowerCAmelCase__  )

 elif numbera == 1 or numbera == 1:
  __a :  Optional[Any]						=  []
  __a :  Optional[Any]						=  []
  __a :  int						=  max(lowerCAmelCase__  ,   lowerCAmelCase__  )

 __a :  Any						=  0
 __a :  str						=  0

 __a :  List[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:
    __a :  List[Any]						=  prime_fac_a.count(lowerCAmelCase__  )
    __a :  Optional[int]						=  prime_fac_a.count(lowerCAmelCase__  )

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

   else:
    __a :  Optional[int]						=  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:
   __a :  List[str]						=  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    __UpperCamelCase     (				lowerCAmelCase__							:				Union[str, Any]  ):
 assert isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and (n >= 0), "'number' must been a positive int"

 __a :  str						=  0
 __a :  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    __UpperCamelCase     (				lowerCAmelCase__							:				Tuple  ,   lowerCAmelCase__							:				Optional[int]  ):
 assert (
     is_prime(lowerCAmelCase__  ) and is_prime(lowerCAmelCase__  ) and (p_number_a < p_number_a)
 ), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

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

 __a :  Dict						=  []  # 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    __UpperCamelCase     (				lowerCAmelCase__							:				Any  ):
 assert isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and (n >= 1), "'n' must been int and >= 1"

 __a :  Tuple						=  []  # 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    __UpperCamelCase     (				lowerCAmelCase__							:				Any  ):
 assert isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and (
     number > 1
 ), "'number' must been an int and >= 1"

 __a :  Tuple						=  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    __UpperCamelCase     (				lowerCAmelCase__							:				List[Any]  ,   lowerCAmelCase__							:				Optional[Any]  ):
 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.
 __a :  str						=  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    __UpperCamelCase     (				lowerCAmelCase__							:				List[str]  ):
 assert isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and (n >= 0), "'n' must been a int and >= 0"

 __a :  Tuple						=  1  # this will be return.

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

 return ans




def    __UpperCamelCase     (				lowerCAmelCase__							:				Union[str, Any]  ):
 assert isinstance(lowerCAmelCase__  ,   lowerCAmelCase__  ) and (n >= 0), "'n' must been an int and >= 0"

 __a :  Any						=  0
 __a :  int						=  1
 __a :  int						=  1  # this will be return

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

 return ans


def    __UpperCamelCase     (				lowerCAmelCase__							:				int = 5_0_0_0_0_0_0_0  ):
 __a :  int						=  set()
 __a :  str						=  int((limit - 2_4) ** (1 / 2)  )

 __a :  int						=  set(range(3  ,   prime_square_limit + 1  ,   2  )  )
 primes.add(2  )
 for p in range(3  ,   prime_square_limit + 1  ,   2  ):
  if p not in primes:
   continue
  primes.difference_update(set(range(p * p  ,   prime_square_limit + 1  ,   lowerCAmelCase__  )  )  )

 for primea in primes:
  __a :  Union[str, Any]						=  primea * primea
  for primea in primes:
   __a :  Union[str, Any]						=  primea * primea * primea
   if square + cube >= limit - 1_6:
    break
   for primea in primes:
    __a :  int						=  primea * primea * primea * primea
    __a :  Union[str, Any]						=  square + cube + tetr
    if total >= limit:
     break
    ret.add(lowerCAmelCase__  )

 return len(lowerCAmelCase__  )


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