def divisors(n: int) -> list[int]:
    xs = set()
    d = 1
    while d * d <= n:
        if n % d == 0:
            xs.add(d)
            xs.add(n // d)
        d += 1
    return sorted(list(xs))


def sigma2(n: int) -> int:
    return sum(d**2 for d in divisors(n))


def sum2(a: int, b: int) -> int:
    return sum(x**2 for x in range(a, b + 1))


def sigma2caps_naive(n: int) -> int:
    return sum(sigma2(i) for i in range(1, n + 1))


def sum2(a: int, b: int) -> int:
    # Calculate sum(x**2 for x in range(a, b + 1)) directly
    # https://stackoverflow.com/questions/70902823/
    r = b * (b + 1) * (2 * b + 1)
    r -= (a - 1) * a * (2 * (a - 1) + 1)
    r //= 6
    return r


def sigma2caps(n: int) -> int:
    res = 0
    mod = 10**9
    a = 1
    while a <= n:
        d = n // a
        b = n // d
        s2 = sum2(a, b) % mod
        res += (s2 * d) % mod
        res %= mod
        a = b + 1
    return res


def euler_401():
    assert sum2(23, 1337) == sum(x**2 for x in range(23, 1337 + 1))
    assert sigma2caps_naive(6) == 113
    assert sigma2caps(6) == 113
    assert sigma2caps(10**6) == 385965077
    assert sigma2caps(10**9) == 73475174
    return sigma2caps(10**15)


if __name__ == "__main__":
    solution = euler_401()
    print(f"e401.py: {solution}")
    assert solution == 281632621