I'm curious if there has been any work done to compute $$\sum_{d|x} \phi(d)^n$$ for integers $n>1$? I've been looking for a while and haven't found anything in the literature. I'm specifically looking to simplify or set lower bounds on this sum. Any help would be appreciated!

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    $\begingroup$ It's necessarily multiplicative, so you just compute it for $x=p^k$ for some prime $p$. This is $$1+(p-1)^n+(p-1)^np^n +\cdots + (p-1)^np^{n(k-1)} = 1+(p-1)^n\frac{p^{nk}-1}{p^n-1}$$ $\endgroup$ – Thomas Andrews Nov 4 '16 at 15:21
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    $\begingroup$ See OEIS sequence A029939 for $n=2$. $n=3$ doesn't seem to be in the OEIS. $\endgroup$ – Robert Israel Nov 4 '16 at 15:28
  • $\begingroup$ I'm guessing the reason you haven't found much is that (1) it isn't hard, and (2) the function hasn't been needed anywhere significant. $\endgroup$ – Thomas Andrews Nov 4 '16 at 15:30
  • $\begingroup$ @ThomasAndrews, thanks! And yeah I assume so as well. I was hoping there was a form that did not rely on the prime decomposition of x, but oh well. $\endgroup$ – j4l3kl24jkl2 Nov 4 '16 at 15:37

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