Length to merge of preliminary pairs based on Septembrino's theorem
Follow up to [Connecting Septembrino's theorem with known tuples : r/Collatz](https://www.reddit.com/r/Collatz/comments/1mzu5is/connecting_septembrinos_theorem_with_known_tuples/).
The theorem states ([Paired sequences p/2p+1, for odd p, theorem : r/Collatz](https://www.reddit.com/r/Collatz/comments/1lias5m/paired_sequences_p2p1_for_odd_p_theorem/)): Let p = k•2^n - 1, where k and n are positive integres, and k is odd. Then p and 2p+1 will merge after n odd steps if either k = 1 mod 4 and n is odd, or k = 3 mod 4 and n is even.
The table below show a small portion of the results, with n (and thus k mod 4) in rows and k in column. The preliminary pairs are not Septembrino's pairs and n counts odd numbers.
The partial trees below confirm that Septembrino's pairs for n=1 iterate only once into an odd number before the merge (2-3 involve the trivial cycle, not mentioned here). The segment colors confirm that the three possible sets of segments are used in turn.
[Updated overview of the project (structured presentation of the posts with comments) : r/Collatz](https://www.reddit.com/r/Collatz/comments/1mf41d3/updated_overview_of_the_project_structured/)
Posted Aug 31, 2025, 1:41 PM UTC