  We embed the somewhat unusual multiplicative function, which was
serendipitously discovered in 2010 during a study of mutually unbiased bases in
the Hilbert space of quantum physics, into two families of multiplicative
functions that we construct as generalizations of that particular example. In
addition, we report yet another multiplicative function, which is also
suggested by that example; it can be used to express the squarefree part of an
integer in terms of an exponential sum.
