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

Paul Pollack ¹   ; Carl Pomerance ²

¹ Department of Mathematics University of Georgia Athens, GA 30602, U.S.A.
² Department of Mathematics Dartmouth College Hanover, NH 03755, U.S.A.

Colloquium Mathematicum, Tome 130 (2013) no. 2, pp. 169-182

Résumé

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 $x/(\log{x})^{1/2+o(1)}$.

Détail

DOI : 10.4064/cm130-2-3

Keywords: sum mid sum nontrivial divisors natural number where nontrivial excludes nbsp example natural number called quasiperfect while said form quasiamicable pair latter called quasiamicable numbers paper prove statistical theorems about these classes numbers first count quasiperfect leq infty each fixed there natural numbers leq sigma equiv nbsp mod sigma odd quasiperfect satisfy these conditions fixed delta neq define arithmetic function delta sigma n delta second theorem says number leq which amicable respect delta log