If you are asking if this is algebra or number theory, I think it's number theory because it has to do with counting numbers. Hope this helps!

This sounds like inverse (but you get to choose, so it's much easier) Collatz. Do you mean $\lfloor x/2 \rfloor$?

If you meant $\lfloor x/2 \rfloor,$ then induction suffices. Suppose $1, 2, \dots, i$ have been reached. If $i = 3k,$ then $3k+1$ is reached from $k.$ If $i = 3k+1,$ then $3k+2$ is reached from $6k+4 = 3(2k+1)+1.$ If $i = 3k-1,$ then $3k$ is reached from $6k+1 = 3(2k)+1.$ This completes the induction by casework.