Originally, I was trying to work out an explicit formula for a function with the property that $f(x + \frac{1}{f(x)}) = f(x) + k$ for some $k$.
This is inspired by a thing I did with a metronome once, where on each beat I turned the frequency dial up by one tick, if that helps you visualize it.
I have computed some points on this curve for $f(0) = 1$ and $k = 1$:
x | f(x) ------+------ 0 | 1 1 | 2 1.5 | 3 1.833 | 4 2.083 | 5 2.283 | 6 2.45 | 7
After computing these points, I realized that this is an inverse function for the n-th partial sum of the harmonic series, or more precisely, $\forall f(x) \in \Bbb Z:\sum_{n=1}^{f(x)} \frac{1}{n} = x$.
What is an explicit formulation for $f(x)$ over the reals, if such exists? (Or if not, why.)