





from __future__ import annotations

from collections.abc import Generator





def lowerCAmelCase__()				->      Generator[int, None, None]:







	'''simple docstring'''







	lowerCamelCase__								=		{}
	lowerCamelCase__								=		2
	while True:
		lowerCamelCase__								=		factor_map.pop(lowerCamelCase_       ,lowerCamelCase_							)
		if factor:
			lowerCamelCase__								=		factor + prime
			while x in factor_map:
				x += factor
			lowerCamelCase__								=		factor
		else:
			lowerCamelCase__								=		prime
			yield prime
		prime += 1





def lowerCAmelCase__(__snake_case = 1E10							)				->      int:







	'''simple docstring'''







	lowerCamelCase__								=		sieve()

	lowerCamelCase__								=		1
	while True:
		lowerCamelCase__								=		next(lowerCamelCase_							)
		if (2 * prime * n) > limit:
			return n
		# Ignore the next prime as the reminder will be 2.
		next(lowerCamelCase_							)
		n += 2


if __name__ == "__main__":
			print(solution())







from __future__ import annotations

from math import ceil, floor, sqrt



def 				UpperCamelCase_(      lowerCamelCase_ = 200_0000 )    ->						int:
    _lowercase       :							list[int]      		=		[0]
    _lowercase       :							int

    for idx in range(1							, ceil(sqrt(target * 2 ) * 1.1 ) ):
        triangle_numbers.append(triangle_numbers[-1] + idx )

    # we want this to be as close as possible to target
    _lowercase       :							int      		=		0
    # the area corresponding to the grid that gives the product closest to target
    _lowercase       :							int      		=		0
    # an estimate of b, using the quadratic formula
    _lowercase       :							float
    # the largest integer less than b_estimate
    _lowercase       :							int
    # the largest integer less than b_estimate
    _lowercase       :							int
    # the triangle number corresponding to b_floor
    _lowercase       :							int
    # the triangle number corresponding to b_ceil
    _lowercase       :							int

    for idx_a, triangle_a in enumerate(triangle_numbers[1:]							, 1 ):
        _lowercase       :							Optional[int]      		=		(-1 + sqrt(1 + 8 * target / triangle_a )) / 2
        _lowercase       :							List[str]      		=		floor(lowerCamelCase_ )
        _lowercase       :							Dict      		=		ceil(lowerCamelCase_ )
        _lowercase       :							List[str]      		=		triangle_numbers[b_floor]
        _lowercase       :							List[str]      		=		triangle_numbers[b_ceil]

        if abs(target - triangle_b_first_guess * triangle_a ) < abs(
            target - best_product ):
            _lowercase       :							Union[str, Any]      		=		triangle_b_first_guess * triangle_a
            _lowercase       :							Union[str, Any]      		=		idx_a * b_floor

        if abs(target - triangle_b_second_guess * triangle_a ) < abs(
            target - best_product ):
            _lowercase       :							Any      		=		triangle_b_second_guess * triangle_a
            _lowercase       :							Optional[Any]      		=		idx_a * b_ceil

    return area


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


