  We propose and discuss recursive formulas for conformally covariant powers
$P_{2N}$ of the Laplacian (GJMS-operators). For locally conformally flat
metrics, these describe the non-constant part of any GJMS-operator as the sum
of a certain linear combination of compositions of lower order GJMS-operators
(primary part) and a second-order operator which is defined by the Schouten
tensor (secondary part). We complete the description of GJMS-operators by
proposing and discussing recursive formulas for their constant terms, i.e., for
Branson's $Q$-curvatures, along similar lines. We confirm the picture in a
number of cases. Full proofs are given for spheres of any dimension and
arbitrary signature. Moreover, we prove formulas of the respective critical
third power $P_6$ in terms of the Yamabe operator $P_2$ and the Paneitz
operator $P_4$, and of a fourth power in terms of $P_2$, $P_4$ and $P_6$. For
general metrics, the latter involves the first two of Graham's extended
obstruction tensors. In full generality, the recursive formulas remain
conjectural. We describe their relation to the theory of residue families and
the associated $Q$-curvature polynomials.
