We investigate the early aftershock activity associated with four moderate earthquakes (Mw 6.6–6.7) that occurred recently in Japan. For each aftershock sequence, we examine continuous high-pass filtered seismograms recorded at seismic stations nearby the main fault to identify as many early events as possible. The magnitude of these events is calibrated using aftershocks that are listed in the earthquake catalog of Japan Meteorological Agency (JMA). The analysis of the aftershock decay rates reveals a power-law time dependence with a scaling exponent close to 1.0 that starts from about one minute from the mainshock. Our results demonstrate that the c-value of the Omori–Utsu law is very small, although a lower bound is not established due to completeness problems in the first minute after the mainshock and statistical fluctuations.