ROSECODE 417

Bonus for p416

Min_25 · Math · 2017-08-04

Let $r_{3} (n) = # {(x, y, z) \in Z^{3} ∣ x^{2} + y^{2} + z^{2} = n}$ .

For example, $r_{3} (0) = 1$ , $r_{3} (1) = 6$ and $r_{3} (100) = 30$ .

Let $S (n, m) = \sum_{k = 0}^{m - 1} r_{3} (n + k)$ .

It can be verified that $S (1, 100) = 4168$ and $S (10^{8}, 100) = 6410310$ .

Find $S (10^{17}, 100)$ .

[My timing: 2.6 seconds (PyPy)]

Note: It would be hard without parigp and some papers.