Powerful amicable numbers

Tom 122 / 2011

Paul Pollack Colloquium Mathematicum 122 (2011), 103-123 MSC: Primary 11A25; Secondary 11N37. DOI: 10.4064/cm122-1-10


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.


  • Paul PollackDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    1409 West Green Street
    Urbana, IL 61801, U.S.A.

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