On the Sum of Reciprocals of Amicable Numbers

TL;DR

This paper investigates the sum of reciprocals of amicable numbers, providing bounds on its value, which was previously known to be a constant but not precisely bounded.

Contribution

It establishes new lower and upper bounds on the sum of reciprocals of amicable numbers, advancing understanding of their aggregate behavior.

Findings

Derived explicit bounds for the sum of reciprocals of amicable numbers

Confirmed the sum converges to a constant as shown by Pomerance

Enhanced previous knowledge with tighter bounds

Abstract

Two numbers $m$ and $n$ are considered amicable if the sum of their proper divisors, $s(n)$ and $s(m)$, satisfy $s(n) = m$ and $s(m) = n$. In 1981, Pomerance showed that the sum of the reciprocals of all such numbers, $P$, is a constant. We obtain both a lower and an upper bound on the value of $P$.