Counting modulo pairs

Let $f(m)$ denote the number of integer pairs $(x,m)$ such that there exists a positive integer $y$ satisfies $x^y\equiv 0(modm)$, where $x$ and $m$ are both positive integers and $1\le x\le m$. For example, f(12) = 2, as two pairs (6, 12) and (12, 12) meet the condition.

Define $S(n)=\sum _{m=1}^{n}f(m)$. You are given S(10) = 16 and S(1000) = 5764.

Find $S({10}^{14})$.

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Thanks to baihacker for the idea.