Summary: Two natural properties of integer sequences are introduced and studied. The first, exact realizability, is the property that the sequence coincides with the number of periodic points under some map. This is shown to impose a strong inner structure on the sequence. The second, realizability in rate, is the property that the sequence asympototically approximates the number of periodic points under some map. In both cases we discuss when a sequence can have that property. For exact realizability, this amounts to examining the range and domain among integer sequences of the paired transformations $###$ that move between an arbitrary sequence of non-negative integers Orb counting the orbits of a map and the sequence Per of periodic points for that map. Several examples from the Encyclopedia of Integer Sequences arise in this work, and a table of sequences from the Encyclopedia known or conjectured to be exactly realizable is given.