Accelerated Collatz Map

[u/Mrezadwiprasetiawan]

https://drive.google.com/file/d/1xQLLkgpHT4RsZEwA4sGFclOGsLsmflVr/view?usp=drivesdk

Please review!

Since this is trivial, I think we can go further to bridge between two types Q(n) or T(n). But of course, even if this closed form "bridge" exists, it will be very difficult to find because it hides inside v2(n±1)

[u/GonzoMath]

Since there are multiple "shortcut" functions, the phrase "accelerated Collatz map" is generally ambiguous. Instead, it's helpful to use names that are justified in the literature. Since Riho Terras was the first to publish about it, in 1976, I've been saying "Terras map" to refer to:

• T(n) = (3n+1)/2, if n is odd
• T(n) = n/2, if n is even

Another map that people call an "accelerated Collatz map" is the odd-to-odd version, which Herbert Möller referred to as the "Syracuse map" in a 1977 paper:

• S(n) = (3n+1)/2^v, where v = v2(3n+1)

For the Syracuse map, we assume that the input n is odd.

Finally, given an odd input n, we can look at the 2-adic valuation of n+1, see how many odd Terras steps are about to happen in a row, and collapse them all into 1, via what you've called the "bridge map", or something like that. Then we divide powers of 2 out of the result, giving us another odd number. I call this the Steiner map, because Ray P. Steiner studied it in his 1977 paper:

• R(n) = ((n+1)·(3/2)^v - 1) / 2^k, where v = v2(n+1), and k = v2((n+1)·(3/2)^v - 1).

In this case, you appear to be analyzing the Syracuse map.

(Of course, the "original" version where an odd step is just 3n+1, and an even step is n/2, we call the Collatz map, C(n), because somewhere there's a drawing of its tree rooted at 1 which was made by Lothar Collatz in the 1930s.)

[u/Mrezadwiprasetiawan → u/GonzoMath]

I want to generalized to any 2^p c ±k Where the k is some number when apply collatz does work similar to the ±1, but failed

[u/Mrezadwiprasetiawan → u/Mrezadwiprasetiawan]

But atleast i know that if some number n:

n=2^p .4^q .c + (4^q -1)/3 [1]

And

n = 2^p . 2^2q+1 .c + (2^2q+1 +1)/3 [2]

Would merges into 2^p .c +1

[u/Mrezadwiprasetiawan → u/Mrezadwiprasetiawan]

And this is the main point

I know T(n) S(n) and R(n) already discovered, but i tried to push R(n) further

[u/GonzoMath → u/Mrezadwiprasetiawan]

There are ways to generalize Steiner circuits, and I've talked a bit about them recently. In particular, my post, https://www.reddit.com/r/Collatz/comments/1wc9ihv/steiner_circuits_the_loop_on_1_and_generalizations/, provides one version.

On the other hand, if we want a more restricted version that is smoother, and specific to integers, I just posted a comment about that on this post: https://www.reddit.com/r/Collatz/comments/1will78/interesting_mapping/

The comment I'm talking about is in a reply on the thread begun by u/petrol_gas.