Recursively divisible numbers

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.