## The Floor's Revenge

### 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 $(\sum_{k=2}^{10}$ `f`_{k}(10^{14})$)$ mod (10^{9}+7).