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April 19, 20260 citationsOpen Access

Collatz-Type Maps with Alternating 3n±1

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MFMasanori Fujii

Key Points

  • This research aims to explore how the block length affects convergence in a Collatz-type map with alternating odd-step rules.
  • Investigated a Collatz-type map with odd steps defined as alternating between 3n + 1 and 3n - 1.
  • Performed computations up to 10 million to analyze convergence patterns.
  • Examined the behavior particularly at block length L = 2.
  • At block length L = 2, around half of the initial values enter a unique 44-cycle.
  • Convergence behavior shows significant variation based on the block length used.

Abstract

We study a Collatz-type map in which the odd-step alternates between 3n + 1 and 3n − 1. Computations up to 107 show that convergence depends strongly on the block length L, with a singular behavior at L = 2, where approximately half of the initial values enter a unique 44-cycle.

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Cite This Study

Masanori Fujii (2026) studied this question.

synapsesocial.com/papers/69e47376010ef96374d8f349https://doi.org/10.5281/zenodo.19632916
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