This question is motivated by the sequence of functions generated by inverse Mellin transforms:
$$\Gamma(z)\Gamma(z-1) \to \frac{2K_1(2\sqrt{z})}{\sqrt{z}}\to e^{\frac{1}{\log z}} \to~ ?$$
Once the last function is found the length of the sequence will be $4$ because there are $4$ total functions.
To complete the sequence what is the inverse Mellin Transform of $f(w)=\exp\big(\frac{1}{\log w}\big)?$
We can set this up as:
$$ f(p)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{f(w)}{p^w}~dw $$
Neither Wolfram Alpha nor Mathematica can calculate it.
I suspect $f(p)$ is some kind of step function.