The first quake, measuring at 6.0 on the Richter scale, occurred at 9:21 pm (1421 GMT) Friday in Bengkulu province and caused panic among residents.
The island suffered extensive damage in June 2000 when a strong quake hit the area, killing 94 people.
The tremor was centred 100 kilometers (62 miles) under the seabed and 44 kilometers northwest of the coastal town of Manna, said Armin Paimin, the head of Bengkulu meteorological office.
"There were no reports of damages or deaths but residents scattered out of their homes when the quake took place," he told AFP by telephone.
The second quake was centred 80 kilometers under the seabed and 75 kilometers northwest of the city of Denpasar, said Fauzi, a coordinator at Indonesia's earthquake center.
It struck at 1:20 am Saturday and was felt in Denpasar and the neighboring towns of Mataram in West Nusa Tenggara island and Banyuwangi in East Java province.
There have been two aftershocks each measuring 4.0 on the Richter scale following the first quake, Fauzi said.
Indonesia sits on the "Pacific Rim of Fire" noted for its volcanic and seismic activity.
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