Please find a closed form for partial sum of a function
$$f(x)=a^{1/x}$$
I want it to be expressed in terms of bounded number of elementary functions and/or well known special functions.
No computer algebra systems I have tried so far could find a satisfactory solution. I believe that the expression can exist in the terms of incomplete Gamma function or its generalizations because indefinite integral of this function can be expressed in terms of incomplete Gamma function:
$$\int f(x) dx= x\sqrt[x]{a}-\operatorname{Ei}\left(\frac{\ln a}{x}\right)\ln a$$