  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.
