On the distribution of some integers related to perfect and amicable numbers

Volume 130 / 2013

Paul Pollack, Carl Pomerance Colloquium Mathematicum 130 (2013), 169-182 MSC: Primary 11A25; Secondary 11N25, 11N37. DOI: 10.4064/cm130-2-3


Let $s'(n) = \sum_{d \mid n,\,1 < d < n} d$ be the sum of the nontrivial divisors of the natural number $n$, where nontrivial excludes both $1$ and $n$. For example, $s'(20)= 2 + 4 + 5 + 10 = 21$. A natural number $n$ is called quasiperfect if $s'(n)=n$, while $n$ and $m$ are said to form a quasiamicable pair if $s'(n)=m$ and $s'(m)=n$; in the latter case, both $n$ and $m$ are called quasiamicable numbers. In this paper, we prove two statistical theorems about these classes of numbers.

First, we show that the count of quasiperfect $n \leq x$ is at most $x^{{1/4}+o(1)}$ as $x\to\infty$. In fact, we show that for each fixed $a$, there are at most $x^{{1/4}+o(1)}$ natural numbers $n \leq x$ with $\sigma(n)\equiv a ({\rm mod}\ n)$ and $\sigma(n)$ odd. (Quasiperfect $n$ satisfy these conditions with $a=1$.) For fixed $\delta \neq 0$, define the arithmetic function $s_{\delta}(n) := \sigma(n)-n-\delta$. Thus, $s_{1} = s'$. Our second theorem says that the number of $n\leq x$ which are amicable with respect to $s_{\delta}$ is at most $x/(\log{x})^{1/2+o(1)}$.


  • Paul PollackDepartment of Mathematics
    University of Georgia
    Athens, GA 30602, U.S.A.
  • Carl PomeranceDepartment of Mathematics
    Dartmouth College
    Hanover, NH 03755, U.S.A.

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