Professor.Membrane
link: https://projecteuler.net/problem=546
Problem 546
Define $$f_{k}(n) = \sum_{i=0}^{n} f_{k}(\lfloor \frac{i}{k} \rfloor)$$, where $f_{k}(0) = 1$ and $\lfloor x \rfloor$ denotes the floor function.
For example $f_{5}(10) = 18$, $f_{7}(100) = 1003$, and $f_{2}(10^3) = 264830889564$.
Find $$\left(\sum_{k = 2}^{10} f_{k}(10^{14})\right) \textrm{ mod } (10^{9}+7).$$
Verification that $f_{5}(10) = 18$:
Note $$\sum_{i = 0}^{10} f_{5}\left(\lfloor \frac{i}{5} \rfloor\right) = 5f_{5}(0) + 5f_{5}(1) + f_{5}(2) = 5(1) + 5(2) + 1(3) = 18$$