  Previously, a formula, incorporating a $5F4$ hypergeometric function, for the
Hilbert-Schmidt-averaged determinantal moments $\left\langle \left\vert
\rho^{PT}\right\vert ^{n}\left\vert \rho\right\vert ^{k}\right\rangle
/\left\langle \left\vert \rho\right\vert ^{k}\right\rangle$ of $4 \times 4$
density-matrices ($\rho$), and their partial transposes ($\rho^{PT}|$) was
applied with $k=0$ to the generalized two-qubit separability-probability
question. The formula can, further, be viewed we note here, as an averaging
over "induced measures in the space of mixed quantum states". The associated
induced-measure separability probabilities ($k =1, 2,\ldots$) are found--{\it
via} a high-precision density approximation procedure--to assume interesting,
relatively simple rational values in the two-re[al]bit ($\alpha =
\frac{1}{2}$), (standard) two-qubit ($\alpha = 1$) and two-quater[nionic]bit
($\alpha =2$) cases. We deduce rather simple companion (rebit, qubit,
quaterbit,\ldots) formulas that successfully reproduce the rational values
assumed for {\it general} $k$. These formulas are observed to share certain
features, possibly allowing them to be incorporated into a single master
formula.
