

def        SCREAMING_SNAKE_CASE__	(  lowerCamelCase__      )  ->					list[int]:

 if num <= 0:
  raise ValueError('Input must be a positive integer'      )

 __lowerCamelCase    :    Optional[Any] 	=			[True] * (num + 1)

 __lowerCamelCase    :    List[Any] 	=			2
 while p * p <= num:
  if primes[p]:
   for i in range(p * p     ,			num + 1     ,			lowerCamelCase__      ):
    __lowerCamelCase    :    Tuple 	=			False
  p += 1

 return [prime for prime in range(2     ,			num + 1      ) if primes[prime]]


if __name__ == "__main__":
     import doctest

     doctest.testmod()

     a		=int(input("""Enter a positive integer: """).strip())
     print(prime_sieve_eratosthenes(user_num))







from collections.abc import Callable
from math import pi, sqrt
from random import uniform
from statistics import mean






def        SCREAMING_SNAKE_CASE__	(  lowerCamelCase__      )  ->					List[Any]:

	# A local function to see if a dot lands in the circle.
	def is_in_circle(lowerCamelCase__     ,			lowerCamelCase__      ) -> bool:
		__lowerCamelCase    :    Dict 	=			sqrt((x**2) + (y**2)      )
		# Our circle has a radius of 1, so a distance
		# greater than 1 would land outside the circle.
		return distance_from_centre <= 1

	# The proportion of guesses that landed in the circle
	__lowerCamelCase    :    Optional[int] 	=			mean(
	    int(is_in_circle(uniform(-1.0     ,			1.0      )     ,			uniform(-1.0     ,			1.0      )      )      )
	    for _ in range(lowerCamelCase__      )      )
	# The ratio of the area for circle to square is pi/4.
	__lowerCamelCase    :    Any 	=			proportion * 4
	print(F"The estimated value of pi is {pi_estimate}"      )
	print(F"The numpy value of pi is {pi}"      )
	print(F"The total error is {abs(pi - pi_estimate      )}"      )






def        SCREAMING_SNAKE_CASE__	(  lowerCamelCase__     ,			lowerCamelCase__     ,			lowerCamelCase__ = 0.0     ,			lowerCamelCase__ = 1.0     ,			)  ->					float:
	return mean(
	    function_to_integrate(uniform(lowerCamelCase__     ,			lowerCamelCase__      )      ) for _ in range(lowerCamelCase__      )      ) * (max_value - min_value)






def        SCREAMING_SNAKE_CASE__	(  lowerCamelCase__     ,			lowerCamelCase__ = 0.0     ,			lowerCamelCase__ = 1.0      )  ->					None:

	def identity_function(lowerCamelCase__      ) -> float:
		return x

	__lowerCamelCase    :    str 	=			area_under_curve_estimator(
	    lowerCamelCase__     ,			lowerCamelCase__     ,			lowerCamelCase__     ,			lowerCamelCase__      )
	__lowerCamelCase    :    int 	=			(max_value * max_value - min_value * min_value) / 2

	print('******************'      )
	print(F"Estimating area under y=x where x varies from {min_value} to {max_value}"      )
	print(F"Estimated value is {estimated_value}"      )
	print(F"Expected value is {expected_value}"      )
	print(F"Total error is {abs(estimated_value - expected_value      )}"      )
	print('******************'      )






def        SCREAMING_SNAKE_CASE__	(  lowerCamelCase__      )  ->					None:

	def function_to_integrate(lowerCamelCase__      ) -> float:
		return sqrt(4.0 - x * x      )

	__lowerCamelCase    :    Any 	=			area_under_curve_estimator(
	    lowerCamelCase__     ,			lowerCamelCase__     ,			0.0     ,			2.0      )

	print('******************'      )
	print('Estimating pi using area_under_curve_estimator'      )
	print(F"Estimated value is {estimated_value}"      )
	print(F"Expected value is {pi}"      )
	print(F"Total error is {abs(estimated_value - pi      )}"      )
	print('******************'      )


if __name__ == "__main__":
					import doctest

					doctest.testmod()




