Powerful amicable numbers

Paul Pollack¹

¹ Department of Mathematics University of Illinois at Urbana-Champaign 1409 West Green Street Urbana, IL 61801, U.S.A.

Colloquium Mathematicum, Tome 122 (2011) no. 1, pp. 103-123

Résumé

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.

Détail

DOI : 10.4064/cm122-1-10

Keywords: sum mid denote sum proper divisors natural number distinct positive integers said form amicable pair called amicable numbers first example amicable pair known already ancients know there infinitely many amicable pairs opposite direction erd showed set amicable numbers has asymptotic density zero ell geq natural number said ell full ell powerful ell divides whenever prime divides shown erd szekeres number ell full leq asymptotically ell ell infty here ell positive constant depending ell each fixed ell set amicable ell full numbers has relative density zero within set ell full numbers