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Russian surnames ending in "Off" are possessive, where "Off" signifies "son." "Off" is a more modern version (began originating in the 19th and early 20th century) of "ov". Most likely, the name is the father's first name + off. Therefore, we can assume that the father's name was or began with Abakum. Sometimes, however, the part preceding the "off" can be named after an animal or profession (as in "Smith's Son" or "Dog's Son"). "Abakumoff" means "son of Abukum," and it probably originated after some guy named "Abakum" who wanted his name to carry on after having some progeny.
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