






'''simple docstring'''





from math import isqrt, loga


def 				snake_case__							(							UpperCAmelCase							:  int       ):
    lowerCAmelCase__   :Optional[int]      							=				[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       ):
                lowerCAmelCase__   :int      							=				False

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


def 				snake_case__							(							UpperCAmelCase							:  int = 8_0_0_8_0_0    ,			UpperCAmelCase							:  int = 8_0_0_8_0_0       ):
    lowerCAmelCase__   :str      							=				degree * loga(UpperCAmelCase       )
    lowerCAmelCase__   :Any      							=				int(UpperCAmelCase       )
    lowerCAmelCase__   :List[Any]      							=				calculate_prime_numbers(UpperCAmelCase       )

    lowerCAmelCase__   :List[str]      							=				0
    lowerCAmelCase__   :List[str]      							=				0
    lowerCAmelCase__   :Optional[Any]      							=				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() = }""")











from __future__ import annotations

import math
from collections.abc import Callable


def 				snake_case__							(							UpperCAmelCase							:  Callable[[int | float], int | float]    ,			UpperCAmelCase							:  int | float    ,			UpperCAmelCase							:  int | float    ,			UpperCAmelCase							:  int = 1_0_0    ,			):
    lowerCAmelCase__   :int      							=				x_start
    lowerCAmelCase__   :int      							=				fnc(UpperCAmelCase       )
    lowerCAmelCase__   :List[Any]      							=				0.0

    for _ in range(UpperCAmelCase       ):
        # Approximates curve as a sequence of linear lines and sums their length
        lowerCAmelCase__   :Dict      							=				(x_end - x_start) / steps + xa
        lowerCAmelCase__   :Optional[Any]      							=				fnc(UpperCAmelCase       )
        length += math.hypot(xa - xa    ,			fxa - fxa       )

        # Increment step
        lowerCAmelCase__   :Union[str, Any]      							=				xa
        lowerCAmelCase__   :str      							=				fxa

    return length


if __name__ == "__main__":


 def 				snake_case__							(							UpperCAmelCase							:  int       ):
     return math.sin(1_0 * x       )

 print("""f(x) = sin(10 * x)""")
 print("""The length of the curve from x = -10 to x = 10 is:""")
 _a    :   Any        =			10
 while i <= 10_0000:
  print(f"""With {i} steps: {line_length(f, -10, 10, i)}""")
  i *= 10



