Summation of differing roots of the same number

Question

This was a problem that appeared on an assignment dealing with runtime analysis, I was able to figure out that the problem ran in $\Theta(n)$ by simply pulling out the first term of the summation but I'm curious if this has a formula like how the geometric series or series of integers does.

$\sum_{i=1}^k n^{(1/i)} = n + \sqrt{n} + \sqrt[3]{n} + $ ... $ + \sqrt[k]{n}$

Answer 1

I only know one case where this kind of summation appears (in statistical thermodynamics).

For large values of $n$

$$S_k(n)=\sum_{i=1}^{k\,n} n^{\frac 1 i}\approx (k+1)\, n$$

Trying for $n=123456$ and $k=5$ the exact value is $741317.$ while $6\times 123456=740736$