  Let $\mathbf{G}$ be the set of all finite or infinite increasing sequences of
positive integers beginning with 1. For a sequence $S=\{s(n)\}, n\geq1,$ from
$\mathbf{G},$ a positive number $N$ is called an exponentially $S$-number
$(N\in E(S)),$ if all exponents in its prime power factorization are in $S.$
The author \cite{2} proved that, for every sequence $S\in \mathbf{G},$ the
sequence of exponentially $S$-numbers has a density $h=h(E(S))\in
[\frac{6}{\pi^2}, 1].$ In this paper we study the set $\{h(E(S)\}$ of all such
densities.

Number Theory