Order-cum-power statistics of a finite group

Definition
Let $$G$$ be a finite group. The order-cum-power statistics of $$G$$ give information about the number of elements of any given order that are powers of a certain kind. More precise formulations are given below.

In all the versions below, we let $$n$$ be the order of $$G$$ and $$D$$ be the set of natural numbers dividing $$n$$.

Doubly cumulative version
In this version, the statistics are given by the following function $$f: D \times D \to \mathbb{N}$$: $$f(d_1,d_2)$$ is the number of elements $$g \in G$$ such that $$g^{d_1}$$ is the identity element and there exists $$h \in G$$ such that $$h^{d_2} = g$$.

Related notions
If two finite groups have the same order-cum-power statistics, we say that they are order-cum-power statistics-equivalent finite groups.

Weaker statistics

 * Stronger than::Order statistics of a finite group
 * Stronger than::Power statistics of a finite group