Exponentially $S$-numbers

Abstract

Let $\mathbf{S}$ be the set of all finite or infinite increasing sequences of positive integers. For a sequence $S=\{s(n)\}, n\geq1,$ from $\mathbf{S},$ let us call a positive number $N$ an exponentially $S$-number $(N\in E(S)),$ if all exponents in its prime power factorization are in $S.$ Let us accept that $1\in E(S).$ We prove that, for every sequence $S\in \mathbf{S}$ with $s(1)=1,$ the exponentially $S$-numbers have a density $h=h(E(S))$ such that $$\sum_{i\leq x,\enskip i\in E(S)} 1 = h(E(S))x+R(x), where R(x) does not depend on $S$ and $h(E(S))=\prod_{p}(1+\sum_{i\geq2}\frac{u(i)-u(i-1)}{p^i}),$ where $u(n)$ is the characteristic function of $S.$