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Problem 546


Problem 546


The Floor’s Revenge

Define $f_k(n)=\sum^{n}_{i=0}f_k(\lfloor \frac{i}{k} \rloor)$ 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 $(\sum^{10}_{k=2}f_k(10^{14})) \text{ mod } (10^9+7)$.


地板的复仇

记$f_k(n)=\sum^{n}_{i=0}f_k(\lfloor \frac{i}{k} \rloor)$,其中$f_k(0)=1$ ,$\lfloor x \rfloor$表示地板函数(下取整函数)。

例如,f5(10) = 18,f7(100) = 1003,以及f2(103) = 264830889564。

求$(\sum^{10}_{k=2}f_k(10^{14})) \text{ mod } (10^9+7)$。