Are primes differentiable?

We note that $f_p(x) = \lfloor n + p \sin x\rfloor$ has a point where it is not differentiable when $\sin x = \dfrac kp$ , where $k$ is an integer and $1 \le k \le p$ . $k=p$ , when $x = \frac \pi 2$ . We also note that $\sin x$ is symmetrical about $\frac \pi 2$ , then the points where $f(x)$ is not differentiable are when:

$x = \sin^{-1} \frac 1p,\ \sin^{-1} \frac 2p,\ \sin^{-1} \frac 3p,\ \cdots \sin^{-1} \frac {p-1}p,\ \frac \pi 2,\ \sin^{-1} \frac {p-1}p,\ \cdots \sin^{-1} \frac 3p,\ \sin^{-1} \frac 2p,\ \sin^{-1} \frac 1p$

There are a total of $2p-1$ points. For $p=17497$ , $2p-1 = 34\boxed{993}$ .