Set of all densities of exponentially S-numbers

TL;DR

This paper investigates the set of densities of exponentially S-numbers, which are positive integers with prime exponents in a specified sequence, and characterizes the range of possible densities for all such sequences.

Contribution

It provides a detailed analysis of the set of all possible densities of exponentially S-numbers for various sequences S, extending previous results on their existence and bounds.

Findings

The densities form a subset of [6/π^2, 1]

The set of densities is studied and characterized

The range of densities is fully described

Abstract

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.