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, 56Ni synthesis should occur in the strong outflows from the accretion disk. The decay of 56Ni is considered to be the possible origin of SN bumps in the subsequent optical afterglows of LGRBs. If 56Ni originates entirely from the outflows, there is competition between the luminosities of LGRBs and those of the corresponding 56Ni 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 (, where Z and Z ⊙ 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 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 56Ni mass for luminous SNe associated with ULGRBs.
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