A fast computation of density of exponentially $S$-numbers

TL;DR

This paper introduces an efficient polynomial formula for computing the density of numbers with prime exponents in a set S, improving the speed of calculations for the density of exponentially S-numbers.

Contribution

It provides an equivalent polynomial expression for log of the density, enabling faster computation of the density of E(S) sets compared to previous methods.

Findings

Derived a polynomial formula for log h(E(S))

Enabled rapid calculation of densities for various sets S

Improved computational efficiency for density estimation

Abstract

The author \cite{4} proved that, for every set $S$ of positive integers containing 1 (finite or infinite) there exists the density $h=h(E(S))$ of the set $E(S)$ of numbers whose prime factorizations contain exponents only from $S,$ and gave an explicit formula for $h(E(S)).$ In this paper we give an equivalent polynomial formula for $\log h(E(S))$ which allows to get a fast calculation of $h(E(S)).$