Powerful amicable numbers
Volume 122 / 2011
Abstract
Let $s(n):=\sum_{d\mid n,\,d < n}
d$ denote the sum of the proper divisors of the natural number
$n$. Two distinct positive integers $n$ and $m$ are said to form
an amicable pair if $s(n)=m$ and $s(m)=n$; in this case,
both $n$ and $m$ are called amicable numbers. The first
example of an amicable pair, known already to the ancients, is
$\{220, 284\}$. We do not know if there are infinitely many
amicable pairs. In the opposite direction, Erdős showed in
1955 that the set of amicable numbers has asymptotic density zero.
Let $\ell \geq 1$. A natural number $n$ is said to be $\ell$-full
(or $\ell$-powerful) if $p^\ell$ divides $n$ whenever the prime $p$ divides $n$. As shown by Erdős and Szekeres in 1935, the number of $\ell$-full $n \leq x$ is asymptotically $c_\ell x^{1/\ell}$, as $x\to\infty$. Here $c_\ell$ is a positive constant depending on $\ell$.
We show that for each fixed $\ell$, the set of amicable
$\ell$-full numbers has relative density zero within the set of
$\ell$-full numbers.
