An interesting representation of the Collatz problem

[Thomas Scheuerle]

Hello,

I hope this isn't too off-topic, as this post is just trying to highlight a topic that may lead to another sequence, but I don't have a specific sequence suggestion here yet.

Recently I became aware of an interesting representation of the Collatz problem.

N = Sum_{r=0..k-1} ( 2^e_r * 3^(r - k) - (2^(n+1)/(k*3^k)) )

This sum has some integer parameters

some n:  0 < n; some k: 0 < k <= n; and k  integer constants e_1, e_2, ..., e_k

These exponents e_... may range from 0 to n-1 and must all

have individually distinct values.

The Collatz conjecture states that N can become any natural number if

n, k and e_0 .. e_k are chosen appropriately.

This representation of the problem is based on the topic

of K-Special 3-Smooth representations. A topic that apparently

has not yet been covered by many entries in OEIS.

The most important sequence regarding this topic seems to be:

A213539

The first thing that strikes me here, and which might also be important for understanding the Collatz problem, is the position of the powers of two in this sequence. Empirically, it seems to be given by A023359. This seems plausible given the nature of the sequences, but I can't yet prove it.

Any ideas even remotely related to this topic are welcome here, it was meant as a starting point of some inspiration.

kindest regards

Thomas Scheuerle