The Floor's Revenge (noch nicht übersetzt)

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, f₅(10) = 18, f₇(100) = 1003, and f₂(10³) = 264830889564.

Find $(\sum_{k=2}^{10}$ f_k(10¹⁴)$)$ mod (10⁹+7).