I am interested in the following function, $f:{\mathrm{\Delta}}_{n}\to {\mathbb{R}}_{+}$, given by

$f(p)=\sum _{k=1}^{n}\frac{{p}_{k}}{\sum _{j=k}^{n}{p}_{j}}.$

We always have $f(p)\le n$, using the lower bound $\sum _{j\ge k}{p}_{j}\ge {p}_{k}$. However I feel this must be a loose bound on the quantity

$\underset{p\in {\mathrm{\Delta}}_{n}}{sup}f(p),$

since it requires that $\sum _{j>k}{p}_{j}=0$ for all $k$ to be met with equality. Hence, I am wondering what the largest $f(p)$ can be when evaluated over the simplex?