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 $(\sum _{k=2}^{10}{f}_k({10}^{14}))mod({10}^9+7)$.