import numpy as np
from numpy import ndarray
from scipy.optimize import Bounds, LinearConstraint, minimize






def        __UpperCAmelCase    (       UpperCAmelCase       )->       float:



   """simple docstring"""







   return np.dot(__UpperCamelCase,		__UpperCamelCase       )






class      __lowercase					:

      def __init__(	self				:      Any							,							*,
          __lowerCamelCase				:      Union[str, Any] = np.inf							,							__lowerCamelCase				:      Optional[int] = "linear"							,							__lowerCamelCase				:      Optional[int] = 0.0							,							)							->		None:


         '''simple docstring'''

         lowercase							=		regularization
         lowercase							=		gamma
         if kernel == "linear":
            lowercase							=		self.__linear
         elif kernel == "rbf":
            if self.gamma == 0:
               raise ValueError('''rbf kernel requires gamma'''   )
            if not isinstance(self.gamma							,							(float, int)   ):
               raise ValueError('''gamma must be float or int'''   )
            if not self.gamma > 0:
               raise ValueError('''gamma must be > 0'''   )
            lowercase							=		self.__rbf
            # in the future, there could be a default value like in sklearn
            # sklear: def_gamma = 1/(n_features * X.var()) (wiki)
            # previously it was 1/(n_features)
         else:
            lowercase							=		f'Unknown kernel: {kernel}'
            raise ValueError(_SCREAMING_SNAKE_CASE   )

      def        __a		(	self				:      List[Any]							,							__lowerCamelCase				:      str							,							__lowerCamelCase				:      Optional[Any]   )							->		float:


         '''simple docstring'''

         return np.dot(_SCREAMING_SNAKE_CASE							,							_SCREAMING_SNAKE_CASE   )

      def        __a		(	self				:      Optional[int]							,							__lowerCamelCase				:      int							,							__lowerCamelCase				:      Dict   )							->		float:


         '''simple docstring'''

         return np.exp(-(self.gamma * norm_squared(vectora - vectora   ))   )

      def        __a		(	self				:      int							,							__lowerCamelCase				:      Tuple							,							__lowerCamelCase				:      Optional[int]   )							->		None:


         '''simple docstring'''

         lowercase							=		observations
         lowercase							=		classes

         # using Wolfe's Dual to calculate w.
         # Primal problem: minimize 1/2*norm_squared(w)
         #   constraint: yn(w . xn + b) >= 1
         #
         # With l a vector
         # Dual problem: maximize sum_n(ln) -
         #       1/2 * sum_n(sum_m(ln*lm*yn*ym*xn . xm))
         #   constraint: self.C >= ln >= 0
         #           and sum_n(ln*yn) = 0
         # Then we get w using w = sum_n(ln*yn*xn)
         # At the end we can get b ~= mean(yn - w . xn)
         #
         # Since we use kernels, we only need l_star to calculate b
         # and to classify observations

         ((lowercase)		,)							=		np.shape(_SCREAMING_SNAKE_CASE   )

         def to_minimize(__lowerCamelCase				:      Dict   ) -> float:
            lowercase							=		0
            ((lowercase)		,)							=		np.shape(_SCREAMING_SNAKE_CASE   )
            for i in range(_SCREAMING_SNAKE_CASE   ):
               for j in range(_SCREAMING_SNAKE_CASE   ):
                  s += (
                      candidate[i]
                      * candidate[j]
                      * classes[i]
                      * classes[j]
                      * self.kernel(observations[i]							,							observations[j]   )
                  )
            return 1 / 2 * s - sum(_SCREAMING_SNAKE_CASE   )

         lowercase							=		LinearConstraint(_SCREAMING_SNAKE_CASE							,							0							,							0   )
         lowercase							=		Bounds(0							,							self.regularization   )

         lowercase							=		minimize(
             _SCREAMING_SNAKE_CASE							,							np.ones(_SCREAMING_SNAKE_CASE   )							,							bounds=_SCREAMING_SNAKE_CASE							,							constraints=[ly_contraint]   ).x
         lowercase							=		l_star

         # calculating mean offset of separation plane to points
         lowercase							=		0
         for i in range(_SCREAMING_SNAKE_CASE   ):
            for j in range(_SCREAMING_SNAKE_CASE   ):
               s += classes[i] - classes[i] * self.optimum[i] * self.kernel(
                   observations[i]							,							observations[j]   )
         lowercase							=		s / n



      def        __a		(	self				:      Any							,							__lowerCamelCase				:      str   )							->		int:


         '''simple docstring'''

         lowercase							=		sum(
             self.optimum[n]
             * self.classes[n]
             * self.kernel(self.observations[n]							,							_SCREAMING_SNAKE_CASE   )
             for n in range(len(self.classes   )   )   )
         return 1 if s + self.offset >= 0 else -1


if __name__ == "__main__":
      import doctest

      doctest.testmod()






from __future__ import annotations

import math






def        __UpperCAmelCase    (       UpperCAmelCase       )->       list[int]:



			"""simple docstring"""







			if num <= 0:
						lowercase							=		f'{num}: Invalid input, please enter a positive integer.'
						raise ValueError(UpperCAmelCase       )

			lowercase							=		[True] * (num + 1)
			lowercase							=		[]
			lowercase							=		2
			lowercase							=		int(math.sqrt(UpperCAmelCase       )       )

			while start <= end:
						# If start is a prime
						if sieve[start] is True:
									prime.append(UpperCAmelCase       )

									# Set multiples of start be False
									for i in range(start * start,		num + 1,		UpperCAmelCase       ):
												if sieve[i] is True:
															lowercase							=		False

						start += 1

			for j in range(end + 1,		num + 1       ):
						if sieve[j] is True:
									prime.append(UpperCAmelCase       )

			return prime


if __name__ == "__main__":
						print(prime_sieve(int(input("Enter a positive integer: ").strip())))





