Recursively divisible numbers

TL;DR

This paper introduces a recursive divisor function, explores its properties, and connects it to known number classifications, revealing new recursively divisible numbers with potential applications.

Contribution

It defines and analyzes the recursive divisor function, providing geometric interpretations and linking it to ordered factorizations, thus offering new insights into number divisibility properties.

Findings

$oldsymbol{ ext{For } n ext{ ≥ 2, } oldsymbol{rac{oldsymbol{ ext{kappa}_x(n)}}{oldsymbol{n^x}} < 1/(2-oldsymbol{ ext{ exteta}}(x))}}$

$oldsymbol{ ext{kappa}_0(n)}$ is twice the number of ordered factorizations of n

Identification of numbers more recursively divisible than all predecessors, with applications in design and technology

Abstract

We introduce and study the recursive divisor function, a recursive analog of the usual divisor function: $\kappa_x(n) = n^x + \sum_{d\lfloor n} \kappa_x(d)$, where the sum is over the proper divisors of $n$. We give a geometrical interpretation of $\kappa_x(n)$, which we use to derive a relation between $\kappa_x(n)$ and $\kappa_0(n)$. For $x \geq 2$, we observe that $\kappa_x(n)/n^x < 1/(2-\zeta(x))$. We show that, for $n \geq 2$, $\kappa_0(n)$ is twice the number of ordered factorizations, a problem much studied in its own right. By computing those numbers that are more recursively divisible than all of their predecessors, we recover many of the numbers prevalent in design and technology, and suggest new ones which have yet to be adopted.