Echoes of Constancy: Waves of Change in the Keçeci and Oresme Sequences

Resumen

This study presents a comparative analysis of static and dynamic number sequences, using the classical Oresme numbers and the novel Keçeci numbers, developed by Mehmet Keçeci, as primary case studies. Static sequences are characterized by a fixed, predictable recurrence relation. The Oresme numbers—the partial sums of the harmonic series (Η_n=∑(k=1)^n 1/k)—exemplify this category. Their generation follows a simple, deterministic rule (Η_n= Η(n-1)+1/n), and their predictable divergence, proven by Nicole Oresme, serves as a foundational concept in mathematical analysis and pedagogy. In stark contrast, Keçeci numbers are defined as a dynamic sequence generated by a state-dependent algorithm. Their progression is not linear but determined by the properties of the terms themselves. The algorithm initiates with a value and an increment, but each subsequent term is derived through a conditional pathway involving division by an alternating divisor (2 or 3). If division fails, a primality check is performed on the term's principal component (e.g., the real part of a complex number). A prime result triggers the unique "Augment/Shrink & Check (ASK)" rule, modifying the term before re-attempting division. This process, implemented in Python for number sets including integers, rationals, complex numbers, and quaternions, generates a complex, path-dependent behaviour. The comparison reveals a fundamental dichotomy. Oresme numbers provide a robust, transparent framework ideal for theoretical exploration and teaching mathematical series. Conversely, the dynamic and adaptive structure of Keçeci numbers offers significant flexibility, suggesting potential applications in modern computational fields such as algorithm design, cryptographic systems, and procedural generation in simulations. While the predictable nature of static sequences like Oresme's provides a solid theoretical bedrock for analysis, the computationally intensive and pseudo-random characteristics of dynamic sequences like Keçeci numbers open new research avenues in computer science and complex systems modelling.