One sum computation

Solving one problem I encountered one problem. I needed calculate one interesting sum: sigma(N / i), where (1 <= i && <= K). How we can do it as fast as possible. We can do it using complexity sqrt(N) moving from two ends. Following code demonstate this approach.

    long smart(long N, long K) {
        long res = 0;
        long t = (long)Math.sqrt(N * 1.0);
        for (long i = 1; i * i < N; ++i) {
            if (i <= K) res += N / i;
            long lo = Math.max(N / (i + 1), t);
            long hi = Math.min(N / i, K);
            if (hi > lo) res += (hi - lo) * i;
        }
        if (t * t == N && t <= K) res += t;
        return res;
    }

After it I recall one neerc problem. But there we need compute a little different sum: sigma(N % i), where (1 <= i && <= K). So we can replace N % i to (N - N / i * i) and addopt previous code to calculate sigma(N / i * i).