The paper investigates the role of Jacobsthal recurrent numbers in forming statistical patterns within the model of the natural number hypothesis q ϵ ℕ in the general problem of the form κ⋅q±1, where κ=1,3,5,…. A novel model is proposed for structuring the set of natural numbers as sequences of the form θ⋅2n, where the parameter θ takes odd values 1,3,5,…, and n is a natural number starting from zero. A branching and merging diagram of such sequences has been developed, describing their evolution towards a general stopping time tst, where tst→∞. The properties of these structures are investigated, particularly their relationship with the dynamics of the Collatz conjecture. Based on the proposed model, the formation of number sequences with the same length in the Collatz conjecture CSq has been identified for the first time. The obtained results can be used for further analysis of arithmetic transformations and properties of natural numbers in the context of number theory.
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