Does the arithmetical function $$ s(n) := \sum_{k|n} \min\left\{k,\frac{n}{k}\right\} $$ have well-studied properties? Can anybody help me by pointing out some studies about these functions?
As motivation, I am studying about the number of solutions of some Diophantine equations and try to come up with some identities involving arithmetic functions. One of the equations gives result involving this function.
Thank you.
Let $d_1,\ldots,d_r$ be the positive divisors of $n$, in increasing order, hence $d_1=1$, $d_r=r$, and $n=d_id_{r+1-i}$ for all $i$. It is well-known that $r$ is odd if and only if $n$ is square. Therefore, if $n$ is a square then $r$ is odd and $$ s(n)=d_{\frac{r+1}{2}}+2\left(d_1+\cdots+d_{\frac{r-1}{2}}\right) $$ Otherwise $n$ is not square, so $n$ is even and $$ s(n)=2\left(d_1+\cdots+d_{\frac{r}{2}}\right). $$
To answer your question, which properties of $s$ you have in mind?