On the Within-perfect Numbers

Abstract

Motivated by the works of Erdös, Pomerance, Wolke and Harman on the sum-of-divisor function $\ddot{I}ƒ(n)$, 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 $k:[1,\infty )\to (0,\infty )$ is an increasing and unbounded function.

Minor changes, Accepted for publication. This article together with arXiv:1610.04253v4 (published in Acta Arithmetica 194 (2020), 341-366) supersede arXiv:1610.04253v2