**EXPANDED ABSTRACT of the paper**

*“Estimating
the counts of Carmichael and Williams numbers with
small multiple seeds***”,**

**Math.**** Comp., 84:291
(2015), 309-337.**

**by**** Zhenxiang Zhang**

Let *b*_{i}* *be the *i*th prime and let

(1)
* L =** *∏_{1 ≤ }_{i }_{≤ }_{s}_{ }*b*_{i}* *^{e}^{i}

with *s **≥ *2, *e*_{i}* **≥ *1 for *i *∈* *{1,* s*}* *and *e _{i}*

Define the set *D* (*L*)
= {*d *∈* *Z^{+} : 2 *| **d*;* ** **d **| **L*}, which is the set of *even** *divisors of *L*.

Define sets
*P** *(*L*)
= {prime *p** *: *p **− *1 ∈* D** *(*L*);* p *∤ *L*};

*N** *(*L*)
={*n* >* *1 : *n** *is square-free and composed entirely of
the primes in *P** *(*L*) };

and

*C*** **(

Then every
number *n** *∈* C** *(*L*) is
Carmichael.

In 1977 Williams asked that whether there are
any Carmichael numbers* n* with an odd number of prime divisors and the additional property that
for each prime *p* | *n*,

(2)* ** **p** *+ 1 |* **n *+ 1.

In 2007 Echi omitted
the phrase “with an odd number of prime divisors” to relax the
Williams question. Echi then said: “*This is a long standing open question**;* *and it is possible that there is no such number*.” So, he extended the
problem of Williams: he called a square-free composite *n** *an *a*-*Williams number *if *p**−**a *|* **n**−**a* and
*p*+*a** **| **n*+*a** *for all primes
*p *dividing *n*, where *a** *is a given nonzero
integer. In 2010 Bouall`egue, et al. found some *a*-Williams numbers *n < *10^{8}, with *a **≈ **n*/3. In contrast to Echi,
we believe that there should be infinitely many Carmichael numbers
satisfying (2). We call such
Carmichael numbers *Williams numbers** *(which are 1-Williams numbers by Echi), though so far
not a single Williams number has been found.

Let *L** *be as defined in (1). Let

*
**L*^{(}^{1)} = 2 ∏_{2 ≤ }_{j }_{≤ }_{s, j }_{even}* **b*_{j}* *^{e}^{j}

and

*L*^{(}^{2)} = ∏_{1 ≤ }_{j }_{≤ }_{s, j }_{odd}_{ }*b*_{j}* *^{e}^{j}

Let

*Q** *(*L*)
={prime *q** *: *q **− *1 ∈* D** *(*L*^{(1)}), * **q *+ 1
∈* D** *(*L*^{(2)});* ** **q *∤* **L*},

*N**’*(*L*) ={*n* >* *1 : *n** *is square-free and composed entirely of
the primes in *Q* (*L*) },

*C**’*(*L*) ={*n *∈* **N**’*(*L*) : *L*^{(1)} *| **n **− *1},

and

*W** *(*L*)
={*n *∈* C**’** *(*L*)
: *L*^{(2)}|* **n *+ 1} = {*n *∈* **N**’** *(*L*)
: *L*^{(1)} |* **n **– *1,* **L*^{(2)}|* **n *+ 1}.

Then every number *n *∈* **W** *(*L*) is Williams. **In this paper**, we evaluate both |*C** *(*L*)|* *and |*W** *(*L*)|* *for general (non-square-free) *L*. We call *L *an *even*** -divisors-optimal number **(

**Conjecture 1. ***For an EDON **L** **with
** **ω*(*L*) = *s**, **we
have** *log_{2} log_{2}*|**C *(*L*)*| *= *s*(1+*o*(1))*.*

**Conjecture 2. ***For an EDON **L** **with
** **ω*(*L*) = *s**, **we
have**
*log_{2} log_{2}*|**W *(*L*)*| *=*s*^{ }^{1}^{/}^{2}^{−}^{o}^{(1)}.

Our reasons for making Conjectures 1 and 2 are mainly based on the
heuristics of ErdÖs,
Alford, Granville, and Pomerance concerning
ErdÖs’s
construction of Carmichael numbers and based on Theorem 1.

**Theorem 1. ***If **L **is an EDON with**ω*(*L*) = *s**, **then we have*

*L < *2 (*b*_{s}_{+1})^{2s}
* **and* |*D* (*L*)| = 2^{s}^{(1+}^{o}^{(1))}.

In **§****3** we prove Theorem **§****4** we describe an
algorithm, based on Theorem 1, to generate EDONs and tabulate some of them with relative values
which are necessary for next sections. In **§****5, we** give numerical evidence to support Conjecture 1.
By an adaptation of
the procedure described in **§****2** of our
2011 paper for
computing *|**C *(*L*)*|** *for a square-free *L** *to compute |*C* (*L*)| for a general non-square-free
*L*, we find 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}^{ }. In **§****6**, we describe procedures
for obtaining numerical evidence of |*Q** *(*L*)|* *to
support Conjecture 2. We exhibit such EDONs *L** *that *W** *(*L*) might
be non-empty. In **§****7**,we tabulate a table
of **1029 ****primes**,
and establish that there are at least **2**** ^{24}** Williams numbers made up from them, thought we
have not yet actually found any Williams number. We state the difficulties for
explicitly finding Williams numbers, and predict what a Williams number
looks like. (

2022-1-18