The aliquot constant

TL;DR

This paper investigates the long-term behavior of the sum of proper divisors of even integers, showing that the average logarithmic growth rate converges to a negative constant, suggesting aliquot sequences may tend to stay bounded.

Contribution

It proves the convergence of the average log s(n)/n for even integers and estimates the constant to be less than zero, providing probabilistic evidence for boundedness of aliquot sequences.

Findings

The average of log s(n)/n converges to a negative constant.

The constant lambda is approximated and shown to be less than 0.

The geometric mean of s(n)/n tends to be less than 1.

Abstract

The average value of log s(n)/n taken over the first N even integers is shown to converge to a constant lambda when N tends to infinity; moreover, the value of this constant is approximated and proven to be less than 0. Here s(n) sums the divisors of n less than n. Thus the geometric mean of s(n)/n, the growth factor of the function s, in the long run tends to be less than 1. This could be interpreted as probabilistic evidence that aliquot sequences tend to remain bounded.