






"""simple docstring"""






from __future__ import annotations

from collections.abc import Generator



def 					_A			() ->       Generator[int, None, None]:
    """simple docstring"""






    SCREAMING_SNAKE_CASE_      :  dict[int, int]    						=		{}
    SCREAMING_SNAKE_CASE_      :  List[Any]    						=		2
    while True:
        SCREAMING_SNAKE_CASE_      :  int    						=		factor_map.pop(__a    ,				__a							)
        if factor:
            SCREAMING_SNAKE_CASE_      :  Union[str, Any]    						=		factor + prime
            while x in factor_map:
                x += factor
            SCREAMING_SNAKE_CASE_      :  List[str]    						=		factor
        else:
            SCREAMING_SNAKE_CASE_      :  List[str]    						=		prime
            yield prime
        prime += 1



def 					_A			(__a = 1e10							) ->       int:
    """simple docstring"""






    SCREAMING_SNAKE_CASE_      :  Tuple    						=		sieve()

    SCREAMING_SNAKE_CASE_      :  Optional[int]    						=		1
    while True:
        SCREAMING_SNAKE_CASE_      :  Union[str, Any]    						=		next(__a							)
        if (2 * prime * n) > limit:
            return n
        # Ignore the next prime as the reminder will be 2.
        next(__a							)
        n += 2


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



"""simple docstring"""
from math import pow, sqrt







def 						__a   (	*_SCREAMING_SNAKE_CASE   )	->bool:
  a__:     Union[str, Any]										=	len(_SCREAMING_SNAKE_CASE   ) > 0 and all(value > 0.0 for value in values   )
  return result







def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )	->float | ValueError:
  return (
      round(sqrt(molar_mass_a / molar_mass_a   )		,     6   )
      if validate(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
      else ValueError('Input Error: Molar mass values must greater than 0.'   )
  )







def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )	->float | ValueError:
  return (
      round(effusion_rate * sqrt(molar_mass_a / molar_mass_a   )		,     6   )
      if validate(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
      else ValueError(
          'Input Error: Molar mass and effusion rate values must greater than 0.'   )
  )







def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )	->float | ValueError:
  return (
      round(effusion_rate / sqrt(molar_mass_a / molar_mass_a   )		,     6   )
      if validate(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
      else ValueError(
          'Input Error: Molar mass and effusion rate values must greater than 0.'   )
  )







def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )	->float | ValueError:
  return (
      round(molar_mass / pow(effusion_rate_a / effusion_rate_a		,     2   )		,     6   )
      if validate(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
      else ValueError(
          'Input Error: Molar mass and effusion rate values must greater than 0.'   )
  )







def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )	->float | ValueError:
  return (
      round(pow(effusion_rate_a / effusion_rate_a		,     2   ) / molar_mass		,     6   )
      if validate(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   )
      else ValueError(
          'Input Error: Molar mass and effusion rate values must greater than 0.'   )
  )


