Security forces in riot gear fired tear gas to block a march on to the presidential palace in the capital, Algiers. But protests in the rest of country passed off peacefully.
The Algerian premier has held office since 1999, but has rarely been seen in public since suffering a stroke in 2013.
The authorities announced on Thursday that the president, who turns 82 next week, will travel to Switzerland on Sunday for medical checks.
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Security forces cordoned off Algiers' May 1 Square during the protest, witnesses told local media.
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Abdelaziz Bouteflika came to power in 1999 despite allegations of election fraud
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Octogenarian Bouteflika is expected to win another election in April despite concerns for his health.
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Algerian demonstrators defaced a parking sign of a man on a wheelchair (representing President Bouteflika, who uses a wheelchair).
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Protests were held several other cities including Oran, Tizi Ouzou, Bejaia, Annaba and Setif, according to news website TSA.