






from math import isqrt, loga






def 					lowerCamelCase__					(_UpperCAmelCase):
			SCREAMING_SNAKE_CASE							=			[True] * max_number
			for i in range(2				,							isqrt(max_number - 1) + 1):
						if is_prime[i]:
									for j in range(i**2				,							_UpperCAmelCase				,							_UpperCAmelCase):
												SCREAMING_SNAKE_CASE							=			False

			return [i for i in range(2				,							_UpperCAmelCase) if is_prime[i]]






def 					lowerCamelCase__					(_UpperCAmelCase = 80_0800				,							_UpperCAmelCase = 80_0800):
			SCREAMING_SNAKE_CASE							=			degree * loga(_UpperCAmelCase)
			SCREAMING_SNAKE_CASE							=			int(_UpperCAmelCase)
			SCREAMING_SNAKE_CASE							=			calculate_prime_numbers(_UpperCAmelCase)

			SCREAMING_SNAKE_CASE							=			0
			SCREAMING_SNAKE_CASE							=			0
			SCREAMING_SNAKE_CASE							=			len(_UpperCAmelCase) - 1
			while left < right:
						while (
						    prime_numbers[right] * loga(prime_numbers[left])
						    + prime_numbers[left] * loga(prime_numbers[right])
						    > upper_bound
						):
									right -= 1
						hybrid_integers_count += right - left
						left += 1

			return hybrid_integers_count


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








"""simple docstring"""
from __future__ import annotations

from numpy import array, cos, cross, floataa, radians, sin
from numpy.typing import NDArray

def 				_snake_case      (   __snake_case							:      float						, __snake_case							:      float						, __snake_case							:      bool = False   ):




   """simple docstring"""







   if radian_mode:
      return [magnitude * cos(__snake_case   ), magnitude * sin(__snake_case   )]
   return [magnitude * cos(radians(__snake_case   )   ), magnitude * sin(radians(__snake_case   )   )]

def 				_snake_case      (   __snake_case							:      NDArray[floataa]						, __snake_case							:      NDArray[floataa]						, __snake_case							:      float = 10**-1   ):




   """simple docstring"""







   _lowerCamelCase	:					NDArray[floataa]   =		cross(__snake_case						, __snake_case   )
   _lowerCamelCase	:					float   =		sum(__snake_case   )
   return abs(__snake_case   ) < eps


if __name__ == "__main__":
 # Test to check if it works
 UpperCAmelCase				=						array(
     [
         polar_force(718.4, 180 - 30),
         polar_force(879.54, 45),
         polar_force(100, -90),
     ]
 )

 UpperCAmelCase				=						array([[0, 0], [0, 0], [0, 0]])

 assert in_static_equilibrium(forces, location)

 # Problem 1 in image_data/2D_problems.jpg
 UpperCAmelCase				=						array(
     [
         polar_force(30 * 9.81, 15),
         polar_force(215, 180 - 45),
         polar_force(264, 90 - 30),
     ]
 )

 UpperCAmelCase				=						array([[0, 0], [0, 0], [0, 0]])

 assert in_static_equilibrium(forces, location)

 # Problem in image_data/2D_problems_1.jpg
 UpperCAmelCase				=						array([[0, -2000], [0, -1200], [0, 1_5600], [0, -1_2400]])

 UpperCAmelCase				=						array([[0, 0], [6, 0], [10, 0], [12, 0]])

 assert in_static_equilibrium(forces, location)

 import doctest

 doctest.testmod()

