Talk:Consistent Collatz

Thank you very much for this interesting contribution !

• Is this wiki entry the first documentation of this work or are there any resources outhere about it (paper, blog, ...)?
• Did you mint the term "consistent Collatz sequence" or was it introduced before in the litterature?
• How do "consistent Collatz sequences" differ from Conway's "generalised Collatz maps"?

Thank you!

I mostly came up with the terminology myself, because I need to call it something in order to be able to write about it. Although I worked out the results on my own, I'm pretty sure other people got there before me; but I'm not sure whether any of them actually wrote down the theory anywhere (as opposed to simply writing programs that made use of it). Collatz functions (which generalize Collatz problems, and which are what Conway studied) already have a lot of documentation (e.g. https://esolangs.org/wiki/Collatz_function); this article is talking about a special case in which all the multipliers are the same, thus "consistent". (This page uses only a single $r$ value for the entire Collatz function; but Collatz functions in general can have different $r$ values for different values of $x\ {\textrm {mod}}\ m$, something which Conway took advantage of when proving the iterated versions to be Turing-complete.) Ais523 (talk) 22:54, 5 August 2024 (UTC)

Some pre-existing research

According to OEIS, the Hydra/Antihydra sequence is (the ceiling of) a multiple of the sequence of powers of 3/2 (although the ratio is hard to compute, and is probably related to the sequence of $j$ in the article) – I suspect that that result generalises to consistent Collatz sequences in general, although I haven't proved it. There are some known results about real multiples of sequences of powers of rationals in this paper and this paper, but unfortunately AFAICT they aren't strong enough to prove any of the results we care about. Ais523 (talk) 14:31, 12 August 2024 (UTC)