$3n+3^k$: New Perspective on Collatz Conjecture

Abstract

Collatz conjecture is generalized to $3n+3^k$ ($k\in N$). Operating as usual, every sequence seems to reach $3^k$ and end up in the loop $3^k,4.3^k,2.3^k,3^k$. The usual $3n+1$ conjecture is recovered for $k=0$. For $k>0$, we noticed the existence of a sequence of period 3, namely, $3^{k-1},2.3^k,3^k$, alongside the cycle $4.3^k,2.3^k,3^k$ encountered in the $3n+1(k=0)$ sequence. A term formula of the $3n+3^k$ conjecture has been derived, and hence the total stopping time.