  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.
