  Long-duration gamma-ray bursts (LGRBs) associated with supernovae (SNe) are
possibly born out of the death of a massive star. After the star collapses, a
stellar-mass black hole (BH) is formed, surrounded by a hyperaccretion disk
with outflows. Blandford-Znajek jets can be launched and then break out from
the envelope to power LGRBs. The jet luminosity depends on the net inflow
accretion rate at the inner radius of the disk. Furthermore, $\rm ^{56}Ni$
synthesis should occur in the strong outflows from the accretion disk. The
decay of $\rm ^{56}Ni$ is considered to be the possible origin of SN bumps in
the subsequent optical afterglows of LGRBs. If $\rm ^{56}Ni$ originates
entirely from the outflows, there is competition between the luminosities of
LGRBs and those of the corresponding $\rm ^{56}Ni$ bumps because of the
material distribution between the disk inflows and outflows. In this paper, we
investigated these two luminosities based on 15 cases of LGRB-SN in the
framework of the BH hyperaccretion inflow-outflow model. Then, one can
constrain the characteristics of the progenitor stars of these LGRBs. The
results indicate that these LGRBs may originate from the low-metallicity
($Z\lesssim 10^{-2}Z_{\odot}$, where $Z$ and $Z_{\odot}$ are the metallicities
of the stars and the Sun, respectively) stars or some massive solar-metallicity
stars. For ultra-LGRBs (ULGRBs), such as GRB 111209A, most of the massive
low-metallicity stars with $Z \lesssim 10^{-2}Z_{\odot}$ could be progenitors
only if very strong outflows are launched from the disks. When the
contributions of nucleosynthesis in the disk outflows are considered, there is
no shortage of $\rm ^{56}Ni$ mass for luminous SNe associated with ULGRBs.
