[Zhang
2015]* Estimating the counts of Carmichael and Williams numbers with small
multiple seeds*, Mathematics of Computation, **84**:291
(2015), 309-337.

ABSTRACT. 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

log_{2}log_{2}*|C(**L*)*| *= *s*(1+*o*(1)) and log_{2}_{ }log_{2}*|W*(*L*)*| *= *s*^{1/2-o(1)}

when such an Even-Divisor Optimal Number (EDON) *L** *has *s** *different prime factors. For example, we
determine that *|C*(735134400)*| **> *2 *・** *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. (**download
the txt file of the 1029 primes**)

(Note that

*|C(**L*)*|***=**243222…969349 (112 digits)

which
seems currently to be *the largest known cardinality *of a set of Carmichael numbers, where

*L* =
735134400 = 2^{6} · 3^{3} · 5^{2} · 7 · 11 · 13 ·
17

is the largest EDON less than 10^{9}.)