Estimating the counts of Carmichael and Williams numbers with small multiple seeds

Abstract:

For a positive even integer $L$, let $\mathcal {P}(L)$ denote the set of primes $p$ for which $p-1$ divides $L$ but $p$ does not divide $L$, let $\mathcal {C}(L)$ denote the set of Carmichael numbers $n$ where $n$ is composed entirely of primes in $\mathcal {P}(L)$ and where $L$ divides $n-1$, and let $\mathcal {W}(L)\subseteq \mathcal {C}(L)$ denote the subset of Williams numbers, which have the additional property that $p+1 \mid n+1$ for each prime $p\mid n$. We study $|\mathcal {C}(L)|$ and $|\mathcal {W}(L)|$ for certain integers $L$. We describe procedures for generating integers $L$ that have more even divisors than any smaller positive integer, and we obtain certain numerical evidence to support the conjectures that $\log _2|\mathcal {C}(L)|=2^{s(1+o(1))}$ and $\log _2|\mathcal {W}(L)|=2^{s^{1/2-o(1)}}$ when such an “even-divisor optimal” integer $L$ has $s$ different prime factors. For example, we determine that $|\mathcal {C}(735134400)| > 2\cdot 10^{111}$. Last, using a heuristic argument, we estimate that more than $2^{24}$ Williams numbers may be manufactured from a particular set of $1029$ primes, although we do not construct any explicit examples, and we describe the difficulties involved in doing so.