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A solution for integers of the form n = 4k + 1 could be given by a set of 2k (+1)s and 2k (−1)s and n itself. (This generalizes the example of 5 given above.) Although not obvious from the definition, the set of amenable numbers is closed under multiplication (the product of two amenable numbers is an amenable number).
In particular, finite direct product of amenable groups are amenable, although infinite products need not be. Direct limits of amenable groups are amenable. In particular, if a group can be written as a directed union of amenable subgroups, then it is amenable. Amenable groups are unitarizable; the converse is an open problem.
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