Recursively abundant and recursively perfect numbers

TL;DR

This paper introduces recursive analogs of abundant and perfect numbers based on a recursive divisor function, revealing parallels and conjectures about their properties and distribution, suggesting deeper links between divisor functions and number classification.

Contribution

The paper defines recursive abundant and perfect numbers using a new recursive divisor function and explores their properties, distributions, and conjectures, extending classical number theory concepts.

Findings

Recursive abundant numbers are either abundant or odd perfect numbers.

Infinitely many pristine (recursively perfect) numbers exist, but they are not odd except for 1.

Conjecture: odd ample numbers not divisible by the first k primes exist, similar to abundant numbers.

Abstract

The divisor function $\sigma(n)$ sums the divisors of $n$. We call $n$ abundant when $\sigma(n) - n > n$ and perfect when $\sigma(n) - n = n$. I recently introduced the recursive divisor function $a(n)$, the recursive analog of the divisor function. It measures the extent to which a number is highly divisible into parts, such that the parts are highly divisible into subparts, so on. Just as the divisor function motivates the abundant and perfect numbers, the recursive divisor function motivates their recursive analogs, which I introduce here. A number is recursively abundant, or ample, if $a(n) > n$ and recursively perfect, or pristine, if $a(n) = n$. There are striking parallels between abundant and perfect numbers and their recursive counterparts. The product of two ample numbers is ample, and ample numbers are either abundant or odd perfect numbers. Odd ample numbers exist but are rare, and I conjecture that there are such numbers not divisible by the first $k$ primes -- which is known to be true for the abundant numbers. There are infinitely many pristine numbers, but that they cannot be odd, apart from 1. Pristine numbers are the product of a power of two and odd prime solutions to certain Diophantine equations, reminiscent of how perfect numbers are the product of a power of two and a Mersenne prime. The parallels between these kinds of numbers hint at deeper links between the divisor function and its recursive analog, worthy of further investigation.