On the Within-perfect Numbers

Motivated by the works of Erdös, 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 `$(\mathrm{\ell };k)$-within-perfect numbers’, where $\mathrm{\ell }>1$ is a real number and $k:[1,\mathrm{\infty })\to (0,\mathrm{\infty })$ is an increasing and unbounded function.