On the Within-perfect Numbers

TL;DR

This paper investigates the distribution of a new class of numbers called '$(ell;k)$-within-perfect numbers', extending the study of perfect numbers by analyzing their relation to the sum-of-divisors function and their distribution properties.

Contribution

It introduces the concept of '$(ell;k)$-within-perfect numbers' and studies their distribution, extending classical work on perfect numbers and the sum-of-divisors function.

Findings

Characterization of '$(ell;k)$-within-perfect numbers'

Results on their distribution patterns

Connections to classical perfect number theory

Abstract

Motivated by the works of Erd\"os, Pomerance, Wolke and Harman on the sum-of-divisor function $\sigma(n)$, we study the distribution of a special class of natural numbers closely related to (multiply) perfect numbers which we term `$(\ell;k)$-within-perfect numbers', where $\ell >1$ is a real number and $k: [1, \infty) \rightarrow (0, \infty)$ is an increasing and unbounded function.