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For a positive even integer L, let P(L) denote the set of primes p for which p − 1 divides L but p does not divide L, let C(L) denote the set of Carmichael numbers n where n is composed entirely of primes in P(L) and where L divides n − 1, and let W(L) ⊆ C(L) denote the subset of Williams numbers, which have the additional property that p + 1 | n + 1 for each prime p | n. We study |C(L)| and |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 log2 |C(L)| = 2s(1+o(1)) and log2 |W(L)| = 2s 1/2−o(1) when such an “even-divisor optimal” integer L has s different prime factors. For example, we determine that |C(735134400)| > 2 · 10111. Last, using a heuristic argument, we estimate that more than 224 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.

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