Recursively divisible numbers
Generalizing the divisor function to find a new kind of number that can be recursively divided into parts, for use in design and technology.
Divisible numbers are useful whenever a whole needs to be divided into equal parts. But sometimes it is necessary to divide the parts into subparts, and subparts into sub-subparts, and so on, in a recursive way. For example, websites are divided into columns for different types of content, each of which is divided into sub-columns of elements or text.
In this project we introduce and study recursively divisible numbers, starting with the recursive divisor function: a recursive analog of the usual divisor function. We give a geometric interpretation of recursively divisible numbers and study the number and sum of the recursive divisors. Just as the divisor function motivates the abundant and perfect numbers, the recursive divisor function motivates their recursive analogs, which we investigate. The striking parallels between these two families of numbers hint at deep links between the usual and recursive divisor function.
By computing the most recursively divisible numbers, we recover many of the grid sizes commonly used in graphic design and digital technologies, and suggest new grid sizes which have yet to be adopted. These are especially relevant to recursively modular systems which operate across multiple organizational length scales.
Recursively divisible numbers are a new kind of number that are highly divisible, whose quotients are highly divisible, and so on.
Parallels between the perfect and abundant numbers and their recursive analogs point to deeper structure in the recursive divisor function.