On a Recursive Integer Sequence Implying the Nonexistence of Odd Perfect Numbers

TL;DR

This paper introduces a recursive integer sequence related to odd perfect numbers, conjectures its convergence to one for all starting values, and classifies cases where this holds, contributing to number theory.

Contribution

It proposes a new recursive sequence linked to odd perfect numbers and proves convergence for a specific class of initial values, advancing understanding in number theory.

Findings

Sequence converges to one for certain initial values

Conjecture that sequence reaches one for all initial values

Classified a family of integers satisfying the conjecture

Abstract

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent term is half the term. In this paper, we conjecture that this sequence eventually reaches one for all initial values. Furthermore, we classify a family of integers for which this conjecture holds.