Patterned Numbers: A Novel Number Classification with Structural and Quantum Algebraic Perspectives

TL;DR

This work defines patterned numbers as base-10 integers for which a small divisor $d\in\{1,\dots,9\}$ appears as a digit in $n$, uniting digit-based and divisor-based perspectives in recreational number theory. It provides a concrete generation criterion, analyzes the first 100 numbers with a density of $0.72$, and develops a detailed taxonomy of two-digit cases, including cases where digits divide the number or the number is divisible by its units. Beyond enumeration, the authors explore rich geometric and dynamical interpretations: patterned numbers give rise to turn operators, patterned dragon curves, and seahorse tessellations, which extend to self-similar tilings and quantum-walk analogies; they also map to DAGs and oscillator networks that encode digit–divisor structure into connectivity and energy landscapes. The framework ties arithmetic structure to geometric, combinatorial, and physical analogies—ranging from domino-like tilings to coupled-oscillator models—suggesting potential cross-pollination with symbolic dynamics, quantum-inspired processes, and graph-based representations. While primarily exploratory and recreational, the results illustrate how simple digit–divisor constraints can generate richly structured, multi-scale objects with potential applications in sequence analysis, operator-style interpretations, and network dynamics.

Abstract

We introduce \emph{patterned numbers}, a digit--divisor-based classification of integers motivated by recreational mathematics. A number is defined to be patterned if at least one of its positive divisors appears as a digit in its base-10 representation. We study the first hundred natural numbers under this definition, analyze frequency and density, compare prime and composite behavior, and propose a generation rule. Visual ``shape diagrams'' along the number line illustrate transitions between patterned numbers. Finally, we comment on potential relevance to sequence-based operators and algebraic intuition in quantum and combinatorial contexts.