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

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.

7 pages, 15 figures