On the number of special numbers

For lack of a better word, a number is called special if it has mutually distinct exponents in its canonical prime factorizaton for all exponents. Let V(x) be the number of special numbers ≤x. We will prove that there is a constant c>1 such that V(x)∼cxlogx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\displaystyle {V(x) \sim \frac {cx}{\log x}}$\end{document}. We will make some remarks on determining the error term at the end. Using the explicit abc conjecture, we will study the existence of 23 consecutive special integers.