
"""simple docstring"""

def      lowercase__    (       lowerCAmelCase					:       int = 1_000_000			)			->   int:






    """simple docstring"""




    UpperCAmelCase          =       [i - 1 for i in range(limit + 1			)]

    for i in range(2			,					limit + 1			):
        if phi[i] == i - 1:
            for j in range(2 * i			,					limit + 1			,					lowerCAmelCase			):
                phi[j] -= phi[j] // i

    return sum(phi[2 : limit + 1]			)


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





"""simple docstring"""

from math import ceil, sqrt




def      lowercase__    (       lowerCAmelCase					:       int = 1_000_000			)			->   int:






    """simple docstring"""




    UpperCAmelCase          =       0

    for outer_width in range(3			,					(limit // 4) + 2			):
        if outer_width**2 > limit:
            UpperCAmelCase          =       max(ceil(sqrt(outer_width**2 - limit			)			)			,					1			)
        else:
            UpperCAmelCase          =       1
        if (outer_width - hole_width_lower_bound) % 2:
            hole_width_lower_bound += 1

        answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1

    return answer


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



