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