Counting Carmichael numbers with small seeds

Let $A_s$ be the product of the first $s$ primes, let $\mathcal{P}_s$ be the set of primes $p$ for which $p-1$ divides $A_s$ but $p$ does not divide $A_s$, and let $\mathcal{C}_s$ be the set of Carmichael numbers $n$ such that $n$ is composed entirely of the primes in $\mathcal{P}_s$ and such that $A_s$ divides $n-1$. Erdős argued that, for any $\varepsilon>0$ and all sufficiently large $x$ (depending on the choice of $\varepsilon$), the set $\mathcal{C}_s$ contains more than $x^{1-\varepsilon}$ Carmichael numbers $\leq x$, where $s$ is the largest number such that the $s$th prime is less than $\ln x^{\varepsilon/4}$. Based on Erdős's original heuristic, though with certain modification, Alford, Granville, and Pomerance proved that there are more than $x^{2/7}$ Carmichael numbers up to $x$, once $x$ is sufficiently large.

The main purpose of this paper is to give numerical evidence to support the following conjecture which shows that $\vert\mathcal{C}_s\vert$ grows rapidly on $s$: $\vert\mathcal{C}_s\vert=2^{2^{s(1-\varepsilon)}}$ with $\lim_{s\rightarrow \infty} \varepsilon=0,$ or, equivalently, $\vert\mathcal{C}_s\vert=A_s^{2^{s(1-\varepsilon')}}$ with $\lim_{s\rightarrow \infty} \varepsilon'=0$. We describe a procedure to compute exact values of $\vert\mathcal{C}_s\vert$ for small $s$. In particular, we find that $\vert\mathcal{C}_9\vert=8,281,366,855,879,527$ with $\varepsilon=0.36393\ldots$ and that $\vert\mathcal{C}_{10}\vert=21,823,464,288,660,480,291,170,614,377,509,316$ with $\varepsilon=0.31662\ldots$. The entire calculation for computing $\vert\mathcal{C}_s\vert$ for $s\leq 10$ took about 1,500 hours on a PC Pentium Dual E2180/2.0GHz with 1.99 GB memory and 36 GB disk space.