"""simple docstring"""


from __future__ import annotations

import matplotlib.pyplot as plt  # type: ignore
import numpy

# initial triangle of Koch snowflake
lowerCAmelCase_:					Optional[Any]     =					numpy.array([0, 0])
lowerCAmelCase_:					Optional[Any]     =					numpy.array([0.5, 0.8_660_254])
lowerCAmelCase_:					List[str]     =					numpy.array([1, 0])
lowerCAmelCase_:					Any     =					[VECTOR_1, VECTOR_2, VECTOR_3, VECTOR_1]







def 					__a   (		A    ,				A      ):






     '''simple docstring'''



     lowercase__				 =							initial_vectors
     for _ in range(A      ):
          lowercase__				 =							iteration_step(A      )
     return vectors







def 					__a   (		A      ):






     '''simple docstring'''



     lowercase__				 =							[]
     for i, start_vector in enumerate(vectors[:-1]      ):
          lowercase__				 =							vectors[i + 1]
          new_vectors.append(A      )
          lowercase__				 =							end_vector - start_vector
          new_vectors.append(start_vector + difference_vector / 3      )
          new_vectors.append(
              start_vector + difference_vector / 3 + rotate(difference_vector / 3    ,				60      )      )
          new_vectors.append(start_vector + difference_vector * 2 / 3      )
     new_vectors.append(vectors[-1]      )
     return new_vectors







def 					__a   (		A    ,				A      ):






     '''simple docstring'''



     lowercase__				 =							numpy.radians(A      )
     lowercase__ ,						lowercase__				 =							numpy.cos(A      ), numpy.sin(A      )
     lowercase__				 =							numpy.array(((c, -s), (s, c))      )
     return numpy.dot(A    ,				A      )







def 					__a   (		A      ):






     '''simple docstring'''



     lowercase__				 =							plt.gca()
     axes.set_aspect("equal"      )

     # matplotlib.pyplot.plot takes a list of all x-coordinates and a list of all
     # y-coordinates as inputs, which are constructed from the vector-list using
     # zip()
     lowercase__ ,						lowercase__				 =							zip(*A      )
     plt.plot(A    ,				A      )
     plt.show()


if __name__ == "__main__":
 import doctest

 doctest.testmod()

 lowerCAmelCase_:					Any     =					iterate(INITIAL_VECTORS, 5)
 plot(processed_vectors)





"""simple docstring"""


from __future__ import annotations

import math







def 					__a   (		A      ):






     '''simple docstring'''



     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


lowerCAmelCase_:					Optional[Any]     =					[num for num in range(3, 1_0_0_0_0_1, 2) if not is_prime(num)]







def 					__a   (		A      ):






     '''simple docstring'''



     if not isinstance(A    ,				A      ):
          raise ValueError("n must be an integer"      )
     if n <= 0:
          raise ValueError("n must be >= 0"      )

     lowercase__				 =							[]
     for num in range(len(A      )      ):
          lowercase__				 =							0
          while 2 * i * i <= odd_composites[num]:
               lowercase__				 =							odd_composites[num] - 2 * i * i
               if is_prime(A      ):
                    break
               i += 1
          else:
               list_nums.append(odd_composites[num]      )
               if len(A      ) == n:
                    return list_nums

     return []







def 					__a   (		):






     '''simple docstring'''



     return compute_nums(1      )[0]


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




