A question regarding Collatz conjecture The Collatz function $T$ is defined on the set $\mathbb{Z}^+$ of positive integers
as: $T(n)=n/2$ if $n$ is even, and $T(n)=3n+1$ if $n$ is odd. Let $T^k$ be the $k$th iteration
of $T$. We say $n$ terminates if $T^k(n)=1$ for some $k$.
Let $n$ be an integer of the form $$3^{2^k(j-1)}+3^{2^k(j-2)}+\cdots +3^{2^k}+1$$
where $k,j\in \mathbb{Z}^+$ and $j$ is odd.
Question: Will $n$ terminate for all such $k,j$?
 A: Not really an answer but there are a few things I'd like to point out:
1) $n_{j,k} = \displaystyle \sum_{i=0}^{j-1} 3^{i \cdot 2^{k}} = \frac{3^{j \cdot 2^{k}}-1}{3^{2^{k}}-1}$
2) Since $j$ is odd, $n_{j,k}$ is also odd.
3) Thus, $T(n_{j,k})=3 \cdot n_{j,k}+1$ and $T^{2}(n_{j,k})=\frac{1}{2}(3 \cdot n_{j,k}+1)$, since $3n+1$ is even for all $n$.
4) $n_{j,k}$ has an interesting form when viewed in base 3. For example,
IntegerDigits[n[7,3],3]
{1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1}

That is, for $n_{j,k}$ we get a 1 followed by $j$ zeros, etc.
5) Here is a plot of the lengths of the sequences $T^{k}(n_{j,k})$ for $1 \le j \le 16$ and $1 \le k \le 8$, although for these sequences I used a highly reduced version of the Collatz function where if $n$ is even, $T(n)=n/2^{\kappa}$, where $\kappa=\max\{k : \, 2^{k}|n \}$:

Even though these numbers have a certain form that would seem to make it easier to prove they have a downward trajectory to 1, after looking around a little bit I really didn't seem to see any patterns. $\{n_{j,k} : j \, \text{odd}, k \in \mathbb{N}^{+}\}$ is a fairly "large" set of integers, so this question might really be similar in difficultly to the full conjecture.
