Can't remember a number theory problem (from Hofstadter?)

Can't remember a number theory problem (from Hofstadter?)

I'm thinking of a problem in number theory in which one applies a
recurrence that's something like doubling $n$ if it's even and taking it
over $3$ if it's odd...but with some ceilings or additions added in. The
conjecture is that every number's orbit eventually gets down to $1$, and I
believe this is unproven. I think there's about a 60% chance I encountered
this problem first in Godel, Escher, Bach by Douglas Hofstadter, where
there might also be some quote about this problem being far beyond the
resources of contemporary mathematics. It also seems like Hofstadter calls
numbers that eventually come down to $1$ "marvelous," "miraculous," or
some such, but Googling such terms got my nowhere.
Does anybody have any idea what problem I'm talking about?