On the density of Sylow numbers

Let $p$ be a prime number. We say that a positive integer $n$ is a Sylow $p$-number if there exists a finite group having exactly $n$ Sylow $p$-subgroups. When $p=2$, every odd integer is a Sylow $2$-number. In contrast, when $p$ is odd, there exist two positive constants $c_p$ and $c_p^{\mathrm{\prime }}$ such that, denoting by $\beta (p,x)$ the number of Sylow $p$-numbers less than or equal to $x$, $$c_{p}x(\log x{)}^{\frac{1}{p-1}-1}\le \beta (p,x)\le c_p^{\mathrm{\prime }}x(\log x{)}^{\frac{1}{p-1}-1}\mathrm{.}$$ Moreover if ${\beta }_s(p,x)$ is the number of positive integers $n\le x$ such that $n$ is the Sylow $p$-number of some finite solvable group then $${\beta }_s(p,x)\sim c_{p}x(\log x{)}^{\frac{1}{p-1}-1}\text{as }x\to \mathrm{\infty }\mathrm{.}$$ In particular, when $p$ is odd, the natural density of Sylow $p$-numbers is $0$.