  Motivated by the recent discovery of superconductivity in the pentalayer
nickelate Nd$_6$Ni$_5$O$_{12}$ [Nature Materials 10.1038], we calculate its
electronic structure and superconducting critical temperature. We find that
electronic correlations are essential for pushing Nd$_6$Ni$_5$O$_{12}$ into the
superconducting doping range as they shift the electron pockets above the Fermi
energy. As a consequence, Nd$_6$Ni$_5$O$_{12}$ can be described with a single
$d_{x^2-y^2}$ orbital per Ni. Instead, for the bilayer nickelate
Nd$_3$Ni$_2$O$_6$ we find correlations to drive the system into a three-orbital
regime also involving the Ni $d_{xz,yz}$ states. We suggest, however, that
single-orbital physics with optimal doping can be restored by substituting 60%
of the trivalent Nd or La by tetravalent Zr.
