Tom, unless it's purchased in NJ. We have to get a permit from the local PD. A 4-part carbon copy form. The top copy goes to the NJ State Police; the yellow goes to local PD; I get the pink and the seller keeps the yellow. I have to believe that both the state & local PD use that for some sort of registry but I'm not 100% certain.
Not trying answer for Tom, but certain states
do indeed have registration but there is none on the Federal level.