For any finite group $G$, let $$\theta(G) := \sum_{g \in G} \frac{o(g)}{\phi(o(g))},$$ where $o(g)$ denotes the order of the element $g$ in $G$, and where $\phi$ is the Euler totient function.

It is not too hard to see that if $G$ is nilpotent, then $\theta(G)$ is in fact equal to $\sigma(|G|)$, i.e. the sum of the divisors of $|G|$. However, it seems that $\theta(G)$ is always less than or equal to $\sigma(|G|)$, and that equality holds if and only if $G$ is nilpotent.

My question is twofold: (1) Is this claim true? (2) What kind of "natural" properties of groups (such as nilpotency) are there that can be checked by only looking at the orders of the elements of the group?