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

TL;DR

This paper introduces 'patterned numbers', a new classification based on digit-divisor relationships, analyzing their distribution, properties, and potential connections to algebraic and quantum concepts.

Contribution

It defines and explores the properties of patterned numbers, including their frequency, distribution, and generation rules, linking recreational math with algebraic and quantum perspectives.

Findings

Patterned numbers appear with certain density among natural numbers.

Prime and composite patterned numbers exhibit distinct behaviors.

Visual diagrams illustrate transitions and structures of patterned numbers.

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.