Yes, in some cases. This stamp is called a "plate number single" because the paper attached to the stamp (called "selvedge") has the code numbers of the printing plates used to print the stamps. There was one plate used to print each color of the design. "A 1 1 1 1 1" means that the first plate in production for each color was used to print this stamp.
Modern stamps with blank selvedge are not usually worth a premium. 19th century stamps with extra paper at the margins are called positional pieces and are worth more than normal stamps. In general you should leave the selvedge attached to the stamp if it has any type of marginal markings on it (plate numbers, arrows, Mail Early in the Day, Mr. Zip, etc).
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