I Is the Expanded Collatz Sequence Unique for Each k Value?

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The discussion centers on the uniqueness of the expanded Collatz sequence for different k values, specifically examining the modification of the sequence by replacing 3n+1 with 3n+2k+1. Participants note that while the original sequence leads to a known cycle (4-2-1), varying k values produce different cycles, such as 3-12-6-3 for k=1. There is uncertainty about whether this expanded sequence has been previously explored, with references to various Wikipedia pages indicating potential overlaps with existing generalizations. The conversation also touches on the complexity of these mathematical problems, echoing sentiments from mathematician Paul Erdös regarding their difficulty. The primary inquiry remains whether the cycles produced by different k values are unique and independent of the initial seed number.
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TL;DR Summary
Expanding the Collatz sequence by replacing 3n+1 with 3n+2k+1 for k=0,1,2,...
For k=0 we get the original sequence, leading to the cycle 4-2-1. If the Collatz conjecture holds, that would be true for all integer values of n>0.
For values of k>0 we get different cycles.
For k=1 we for instance get the cycle 3-12-6-3
What we want to investigate is:
- What cycle is the sequence iterating to for different values of k?
- Is that cycle unique for k (independent of the seed number)?
Has this expanded Collatz sequence been explored previously?
 
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fresh_42 said:
I haven't seen this one. But even Wikipedia lists so many generalizations
https://it.wikipedia.org/wiki/Congettura_di_Collatz
https://de.wikipedia.org/wiki/Collatz-Problem
https://en.wikipedia.org/wiki/Collatz_conjecture
that yours might have been among them. Compare especially the Syracuse function.

I am with Erdös:
I think my expanded form of the Collatz sequence is already covered as a special case of the natural generalization of the Collatz sequence, explored by Conway.
 
elcaro said:
TL;DR Summary: Expanding the Collatz sequence by replacing 3n+1 with 3n+2k+1 for k=0,1,2,...
For k=0 we get the original sequence, leading to the cycle 4-2-1. If the Collatz conjecture holds, that would be true for all integer values of n>0.
For values of k>0 we get different cycles.
For k=1 we for instance get the cycle 3-12-6-3
What we want to investigate is:
- What cycle is the sequence iterating to for different values of k?
- Is that cycle unique for k (independent of the seed number)?

Has this expanded Collatz sequence been explored previously?
If you want to have more information about the cycle(s) of Collatz, watch this short video on Youtube:
 

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