Looking for an infinite series resembling the exponential function

Question

I'm looking for a function $f(x)$ that has the following property:

$\sum_{x=1}^\infty f(kx) = r^k$

for some real $0 < r < 1$, and at least for strictly positive integer $k$.

Does such a function exist?

This could also be thought of in terms of some sequence of real numbers $f[n]$.

Answer 1

We have that $\;r^k=e^{k\log r}\;$ , so

$$e^{k\log r}=\sum_{n=0}^\infty\frac{k^n(\log r)^n}{n!}$$

Yet the above doesn't have the form you seem to be searching...