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A long GRB is thought to occur when a massive star ends its life and thus it is expected to be accompanied by a type Ic supernova (SN) which also results from a massive star. Several nearby long GRBs are observed with type Ic SNe. However, some nearby long GRBs are observed without SNe, even though they are close enough to be able to observe the accompanied SNe (e.g., GRB060614). Within our current picture of long GRBs, long GRBs should be accompanied by SNe and these long GRBs without SNe are one of the challenges for the current theory of long GRBs. One possible reason is that the accompanied SNe were so faint that we just missed them. For example, SN 2008ha was only as bright as ∼-14 mag in the R band at maximum brightness. The explosion energy, ejecta mass, and 56Ni mass of SN 2008ha are estimated as about 1048 erg, 0.1 M⊙, and 0.003 M⊙, respectively. If such a SN is accompanied by long GRBs, we might miss it. We performed numerical calculations of hydrodynamics and radiation transport and confirmed that SN 2008ha could have been a type Ic supernova and originated from a core-collapse SN with a significant amount of fallback. This demonstrates that some type Ic SNe like SN 2008ha could be very faint because of fallback. Our results suggest that long GRBs without SNe could have been accompanied by such faint SNe with fallback and we could have just missed the accompanied SNe and the current picture of long GRBs would be consistent with long GRBs without SNe. [ABSTRACT FROM AUTHOR]