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\begin{center}
\vskip 1cm{\LARGE\bf
On Almost Lehmer Numbers
}
\vskip 1cm
\large
Tomohiro Yamada \\
Center for Japanese Language and Culture\\
Osaka University\\
562-8558, 8-1-1, Aomatanihigashi\\
Minoo, Osaka\\
Japan\\
\href{mailto:tyamada1093@gmail.com}{\tt tyamada1093@gmail.com} \\
\end{center}

\vskip .2 in

\begin{abstract}
We consider composite numbers $n$ such that $\vph(n)$ divides $\ell(n-1)$
for some squarefree divisor $\ell$ of $n-1$.
We discuss two cases, according to whether the number of prime factors of $\ell$ is bounded or not.
We give a few instances and upper bounds for the number of such integers below a given number.
\end{abstract}

\section{Introduction}\label{intro}

Let $\vph(n)$ denote the Euler totient function of $n$.
Clearly, $\vph(p)=p-1$ for any prime $p$.
Lehmer~\cite{Leh} conjectured that there exists no composite number $n$ such that
$\vph(n)$ divides $n-1$ and showed that such an integer must be an odd squarefree integer
with at least seven prime factors.
In other words, if $\vph(n)\mid (n-1)$ and $n$ is composite, then $n$ is odd and
$\omega(n)=\Omega(n)\geq 7$, where $\omega(n)$ and $\Omega(n)$ respectively
denote the number of distinct and not necessarily distinct prime factors of $n$.

For such an integer $n$, Cohen and Hagis~\cite{CH} showed that $\omega(n)\geq 14$ and $n>10^{20}$,
Renze's notebook~\cite{Ren} shows that $\omega(n)\geq 15$ and $n>10^{26}$,
and Pinch claims that $n>10^{30}$ at his research page~\cite{Pin}.
Pomerance~\cite{Pom} showed that the number of such an integer $n\leq x$ is $O(x^{1/2}\log^{3/4} x)$
and $n\leq r^{2^r}$ if $2\leq \omega(n)\leq r$ additionally.
Luca and Pomerance~\cite{LP} showed that the number of such an integer $n\leq x$ is at most
\[\frac{x^{1/2}}{\log^{1/2+o(1)} x}.\]
Furthermore,
Burek and \.{Z}mija~\cite{BZ} showed that $n\leq 2^{2^r}-2^{2^{r-1}}$ if $\vph(n)$ divides $n-1$
and $2\leq \omega(n)\leq r$.

Weakening the condition $\vph(n)\mid (n-1)$,
Grau and Oller-Marc\'{e}n~\cite{GOM} introduced the $k$-Lehmer property that $\vph(n)\mid (n-1)^k$
and called a composite number with this property to be a $k$-Lehmer number.
The first few $2$-Lehmer numbers are $561, 1105, 1729, 2465, \ldots$ (sequence \seqnum{A173703}).
McNew~\cite{McN} showed that for each $k$, the number of $k$-Lehmer numbers is $O(x^{1-1/(4k-1)})$
and the number of integers which are $k$-Lehmer numbers for some $k$ is
at most $x\exp(-(1+o(1))\log x\log\log\log x/\log\log x)$.
McNew and Wright~\cite{MNW} showed that for each $k\geq 3$, there exist at least
$x^{1/(k-1)+o(1)}$ integers $n\leq x$ which are $k$-Lehmer but not $(k-1)$-Lehmer numbers.

In this paper, we would like to discuss intermediate properties
between the $1$-Lehmer (that is, ordinary Lehmer) property and $2$-Lehmer property.

We call a composite number $n$ to be an almost Lehmer number if $\vph(n)$ divides $\ell(n-1)$
for some squarefree divisor $\ell$ of $n-1$
and an $r$-nearly Lehmer number if $\vph(n)$ divides $\ell (n-1)$
for some squarefree divisor $\ell$ of $n-1$ with $\omega(\ell)\leq r$.
The ordinary Lehmer property is equivalent to the $0$-nearly Lehmer property
and an almost Lehmer number can be called an $\infty$-nearly Lehmer number.

The first few almost Lehmer numbers are
\[1729, 12801, 247105, 1224721, 2704801, 5079361, 8355841, \ldots,\]
given in \seqnum{A337316}.
There exist exactly $38$ almost Lehmer numbers below $2^{32}$.
There exist only five $1$-nearly Lehmer numbers $1729$, $12801$, $5079361$, $34479361$,
and $3069196417$ below $2^{32}$ as given in \seqnum{A338998}.

For $r=1, 2, \ldots, \infty$, let $U_r$ be the set of composite
numbers $n$ for which
$\vph(n)$ divides $\ell(n-1)$ for some squarefree divisor $\ell$ of $n-1$ with $\omega(\ell)\leq r$.
Thus, $U_\infty$ denotes the set of almost Lehmer numbers.
We also use the general notion that $S(x)=\{n\leq x, n\in S\}$
denote the set of integers $S$ up to $x$ for a set $S$ of positive integers.
Then McNew's upper bound for $2$-Lehmer numbers immediately yields that
$\#U_r(x)\leq \#U_\infty(x)=O(x^{6/7})$.
The purpose of this paper is to give stronger upper bounds for $\#U_r(x)$ and $\#U_\infty(x)$.
\begin{theorem}\label{th11}
Let $a_r$ be the number of partitions of the multiset 
$\{1, 1, 2, 2, \ldots, r, r\}$
of $r$ integers repeated twice.
Then, there exist two absolute constants $c$ and $c_1$ such that for each integer $r\geq 1$,
\begin{equation}\label{eq11}
\#U_r(x)<c a_r (x\log x)^{2/3} (c_1 \log\log x)^{2r+2/3}.
\end{equation}
Moreover, we have
\begin{equation}\label{eq12}
\#U_\infty(x)<x^{4/5}\exp\left(\left(\frac{4}{5}+o(1)\right)\frac{\log x\log\log\log x}{\log\log x}\right),
\end{equation}
where $o(1)\rightarrow 0$ ad $x\rightarrow\infty$.
\end{theorem}

The first few terms of the sequence $(a_r)$ are
$2, 9, 66, 712, 10457, \ldots$ given in \seqnum{A020555}.
Bender's asymptotic formula~\cite[Theorem 1]{Ben} yields that
\begin{equation}
\log a_r<2r\left(\log (2r)-\log\log(2r)-1-\frac{\log 2}{2}+o(1)\right)
\end{equation}
as $r$ grows.
Hence, we obtain the following estimates.
\begin{corollary}
Setting $c$ and $c_1$ as in Theorem~\ref{th11}, we have
\begin{equation}
\#U_1(x)<2c(x\log x)^{2/3} (c_1 \log\log x)^{2r+2/3}
\end{equation}
and
\begin{equation}
\#U_r(x)<\left(\frac{(e\sqrt{2}+o_r(1))r}{\log r}\right)^{2r} (x\log x)^{2/3} (c_1 \log\log x)^{2r+2/3},
\end{equation}
where $o_r(1)$ tends to zero as $r$ tends to infinity.
\end{corollary}

Our estimates depend on numbers of multiplicative partitions of integers,
which will be discussed in the next section.
Thus, fast growth of $a_r$ prevents us from showing that $\#U_\infty(x)<x^{{2/3}+o(1)}$.

On the other hand, the above instances lead us to conjecture
that there exist infinitely many almost Lehmer numbers.
Moreover, there may be infinitely many $1$-nearly Lehmer numbers,
although such integers are distributed very rarely below our search limit.
However, these also seem to be difficult to prove or disprove;
it is even not known whether there exist infinitely many $2$-Lehmer numbers or not!

\section{Preliminary estimates}

Let $\tau(s)$ be the number of multiplicative partitions of
$s=s_1 s_2 \cdots s_r$ with $s_1\leq s_2\leq \cdots \leq s_r$.
The values of $\tau(s)$ for positive integers $s$ are given in \seqnum{A001055}.

\begin{lemma}\label{lm1}
For each integer $s\geq 1$, let $S(s; x)$ denote the set of positive integers $n\leq x$
such that $s$ divides $\vph(n)$.
Then
\begin{equation}
\#S(s; x)\leq \frac{\tau(s)x(c_1 \log\log x)^{\Omega(s)}}{s},
\end{equation}
where $c_1$ is an absolute constant.
\end{lemma}
\begin{proof}
We observe that if $s\mid \vph(n)$, then
$q_1^{f_1} q_2^{f_2} \cdots q_t^{f_t} q_{t+1} \cdots q_r \mid n$
for some integers $f_1, f_2, \ldots, f_t\geq 2$ and distinct primes $q_1, q_2, \ldots, q_r$
such that 
\[s\mid q_1^{f_1-1} q_2^{f_2-1} \cdots q_t^{f_t-1} (q_1-1)(q_2-1)\cdots (q_r-1).\]
Moreover, we can take such $q_i$'s in the way that there exists a factorization of $s=s_1 s_2 \cdots s_{r+1}$ with $1<s_1\leq s_2\leq \cdots \leq s_r$
such that $q_i\equiv 1\mathmod{s_i}$ for $i=1, 2, \ldots, r$ and $s_{r+1}$ divides $q_1^{f_1-1} q_2^{f_2-1} \cdots q_t^{f_t-1}$.

For each factorization $s=s_1 s_2 \cdots s_{r+1}$, the number of such integers $n\leq x$ does not exceed
\[\sum_{\substack{q_i\leq x, \\ q_i\equiv 1\mathmod{s_i} (i=1, 2, \ldots, r)}} \frac{x}{q_1 q_2 \cdots q_r s_{r+1}}
=\frac{x}{s_{r+1}} \prod_{i=1}^r\left(\sum_{\substack{q_i\leq x, \\ q_i\equiv 1\mathmod{s_i}}}\frac{1}{q_i}\right).\]
We obtain from Erd\H{o}s, Granville, Pomerance, and Spiro~\cite[(3.1)]{EGPS} that for $i=1, 2, \ldots, r$,
\begin{equation}
\sum_{\substack{q_i\leq x, \\ q_i\equiv 1\mathmod{s_i}}}\frac{1}{q_i}<\frac{c_1 \log\log x}{s_i}
\end{equation}
with some absolute constant $c_1$.
Thus, we conclude that the number of integers $n\leq x$ such that $s$ divides $\vph(n)$ corresponding to each factorization
$s=s_1 s_2 \cdots s_{r+1}$ can be bounded from above by
\[\frac{x(c_1 \log\log x)^r}{s_1 s_2 \cdots s_r s_{r+1}}=\frac{x(c_1 \log\log x)^r}{s}.\]
Now the lemma immediately follows noting that $r\leq \Omega(s)$.
\end{proof}

We must note that although $\tau(s)$ is relatively small when $\Omega(s)$ is small but not when $\Omega(s)$ is large.
Indeed, Canfield, Erd\H{o}s, and Pomerance~\cite{CEP} showed that
$\tau(s)=s\exp(-(1+o(1))\log s\log\log\log s/\log\log s)$ for highly factorable integers $s$, which are given in \seqnum{A033833}.
So that, the above lemma cannot be used in order to bound the number of integers $n$ such that
$\vph(n)$ are multiples of $s$ for an arbitrary integer $s$.
Nevertheless, we can show the following upper bound for a certain sum involving $\tau(s)$.

\begin{lemma}\label{lm2}
As $x$ tends to infinity, we have
\begin{equation}\label{eq22}
\sum_{s\leq x}\frac{\tau(s)}{s}<\frac{(1+o(1))e^{2\sqrt{\log x}}\log^{1/4} x}{2\sqrt{\pi}}.
\end{equation}
\end{lemma}
\begin{proof}
Oppenheim~\cite{Opp} proved that
\begin{equation}
\sum_{s\leq x}\tau(s)=\frac{(1+o(1))x e^{2\sqrt{\log x}}}{2\sqrt{\pi} \log^{3/4} x}.
\end{equation}
By partial summation, we immediately obtain \eqref{eq22}.
\end{proof}

\section{Proof of the theorem}
Let $r$ be a positive integer or $\infty$, $x$ denotes a sufficiently large real number,
and $n\leq x$ be an $r$-nearly Lehmer number.
In this section, the implied constants in $\ll$ and the $O$-symbols are absolute
and each $o(1)$ tends to zero as $x$ goes to infinity.

We begin by writing $(n-1)/\vph(n)=k/\ell$,
where $k$ and $\ell$ are coprime integers and $\ell$ is a squarefree integer
with at most $r$ distinct prime factors dividing $n-1$.
We note that $n$ must be odd and squarefree since $\vph(n)$ and $n$ are coprime and $n$ is composite.

Take an arbitrary divisor $d$ of $n$ and write $n=md$.
Since $n$ is squarefree, we have $\ell(md-1)=k\vph(n)=k\vph(m)\vph(d)$
and
\begin{equation}
md\equiv 1 \mathmod{\frac{\vph(d)}{\ell_0}},
\end{equation}
where $\ell_0=\gcd(\ell, \vph(d))$.

It is clear that $\ell_0\mid \ell\mid (n-1)$
and therefore both $\vph(d)/\ell_0$ and $\ell_0$ divide $md-1$.
Let $a\mid\mid b$ denote that $a\mid b$ and $\gcd(a, b/a)=1$.
We observe that if $p^e\mid\mid\vph(d)$, then $p^{e-1}\mid \vph(d)/\ell_0\mid (md-1)$
and if $p\mid\mid\vph(d)$, then $p\mid \vph(d)\mid \ell(n-1)\mid (n-1)^2=(md-1)^2$ and
therefore $p\mid (md-1)$.
Hence, decomposing $\ell_0=\ell_1\ell_2$, where
each prime factor $p$ of $\ell_0$ divides $\ell_1$ if and only if $p\mid\mid\vph(d)$,
we obtain
\begin{equation}\label{eq31}
md\equiv 1\mathmod{\frac{\vph(d)}{\ell_2}}.
\end{equation}

Now let $L_1>x^{1/3}$ and $L_2=L_1^2$ be real numbers which will be chosen later
in different manners according to whether $r$ is an integer or $r=\infty$.
We can easily see that $n$ cannot have a prime factor $p>L_2$.
If $n=mp$ with $p>L_2$, then the above observation yields that $mp\equiv 1\mathmod{(p-1)/\ell_2}$.
Since $p\equiv 1\mathmod{(p-1)/\ell_2}$ clearly, we have $m\equiv 1\mathmod{(p-1)/\ell_2}$
and therefore $m\geq (p-1)/\ell_2$.
However, we see that $p\equiv 1\mathmod{\ell_2^2}$ since $\ell_2^2\mid\vph(p)=p-1$.
Thus, we must have $p<m^2=(n/p)^2<(x/p)^2$ and $p<x^{2/3}\leq L_2$,
which is a contradiction.

Hence, $n$ must have a prime factor $p\leq L_2$.
If $n\geq L_1$ and $n$ has no prime divisor $p\geq L_1$, then the smallest divisor $d\geq L_1$ of $n$
must satisfy $L_1\leq d\leq L_1^2=L_2$.
Clearly, if $n$ has a prime factor $p$ in the range $L_1\leq d\leq L_2$,
then $n$ has a divisor $d=p$ with $L_1\leq d\leq L_2$.
Thus, we observe that $n$ has a divisor $d$ in the range $L_1\leq d\leq L_2$ if $n\geq L_1$.

For each $d$, the number of integers $n=md\leq x$ satisfying \eqref{eq31} is at most
$1+\floor{\ell_2 x/(d\vph(d))}$.
We note that $\ell_2\leq\sqrt{\vph(d)}\leq L_1$.
Hence, using the inequality $d/\vph(d)\ll \log\log d\leq \log\log x$,
which follows from Theorem~328 of Hardy and Wright~\cite{HW}, we have
\begin{equation}\label{eq32}
\begin{split}
\#U_r(x)& \leq L_1+\sum_{\ell_2\leq L_1} \sum_{\substack{L_1\leq d\leq L_2, \\\ell_2^2\mid\vph(d)}}\left(1+\frac{\ell_2 x}{d\vph(d)}\right) \\
& \ll\sum_{\ell_2\leq L_1}\left(\#S(\ell_2^2; L_2)+\sum_{\substack{L_1\leq d\leq L_2, \\\ell_2^2\mid\vph(d)}} \frac{\ell_2 x\log\log x}{d^2}\right).
\end{split}
\end{equation}

Let us estimate $\#U_r(x)$ for $r<\infty$.
Recalling the definition of $a_r$, it is clear that $\tau(s^2)=a_{\omega(s)}$ for any squarefree integer $s$.
Thus, we have $\tau(\ell_2^2)\leq \tau(\ell^2)\leq a_r$.
Using Lemma~\ref{lm1} and partial summation, we obtain
\begin{equation}
\begin{split}
\#U_r(x)& \ll a_r \sum_{\ell_2\leq L_1} \left(\frac{L_2(c_1 \log\log x)^{\Omega(\ell_2^2)}}{\ell_2^2}
+\frac{x(c_1 \log\log x)^{\Omega(\ell_2^2)+1}}{L_1\ell_2}\right) \\
& \ll a_r \left(L_2(c_1 \log\log x)^{2r}+\frac{x(\log x)(c_1 \log\log x)^{2r+1}}{L_1}\right).
\end{split}
\end{equation}
Taking $L_1=(c_1 x\log x\log\log x)^{1/3}$, we obtain the theorem.

Finally, we shall estimate $\#U_\infty(x)$.
Since $\ell_2^2\mid \vph(d)$, we have $\vph(d)/\ell_2\geq\sqrt{\vph(d)}\gg (d/\log\log d)^{1/2}$
using Theorem~328 of Hardy and Wright~\cite{HW} again.
Now, instead of the bottom line of \eqref{eq32}, we obtain
\begin{equation}\label{eq33}
\begin{split}
\#U_\infty(x)\ll & \sum_{\ell_2<L_1}\left(\#S(\ell_2^2; L_2)
+\sum_{\substack{L_1\leq d\leq L_2, \\\ell_2^2\mid\vph(d)}} \frac{x(\log\log x)^{1/2}}{d^{3/2}}\right) \\
\ll & \sum_{\ell_2\leq L_1}\frac{\tau(\ell_2^2)}{\ell_2^2}\left(L_2(c_1 \log\log x)^{\Omega(\ell_2)}
+\frac{x(c_1 \log\log x)^{\Omega(\ell_2)+1/2}}{L_1^{1/2}}\right).
\end{split}
\end{equation}

Since $\ell_2$ is squarefree, we have $\Omega(\ell_2^2)=2\omega(\ell_2)$.
Hence, from Hardy and Wright~\cite[Chapter 22.10]{HW}, we see that
\begin{equation}\label{eq34}
\Omega(\ell_2^2)<\frac{2(1+o_{\ell_2}(1))\log\ell_2}{\log\log\ell_2}<\frac{(1+o(1))\log L_2}{\log\log x},
\end{equation}
where the former $o_{\ell_2}(1)$ tends to zero as $\ell_2$ goes to infinity
but the latter $o(1)$ tends to zero as $L_2$ (and therefore $x$) goes to infinity.
By Lemma~\ref{lm2}, we have
\begin{equation}\label{eq35}
\sum_{\ell_2<L_1}\frac{\tau(\ell_2^2)}{\ell_2^2}\leq\sum_{s<L_2}\frac{\tau(s)}{s}\ll e^{2\sqrt{\log x}}\log^{1/4} x.
\end{equation}
Inserting \eqref{eq34} and \eqref{eq35} into \eqref{eq33}, we obtain
\begin{equation}
\#U_\infty(x)\ll e^{(1+o(1))\log L_2 \log\log\log x/\log\log x}\left(L_2+\frac{x}{L_1^{1/2}}\right).
\end{equation}
Now the theorem immediately follows taking $L_1=x^{2/5}$.
This completes the proof.

\begin{thebibliography}{99}
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\bibitem{BZ}
Dominik Burek and B{\l}a\.{z}ej \.{Z}mija,
A new upper bound for numbers with the Lehmer property
and its application to repunit numbers,
{\it Int. J. Number Theory} {\bf 15} (2016), 1463--1468.

\bibitem{CEP}
E. R. Canfield, P. Erd\H{o}s, and C. Pomerance,
On a problem of Oppenheim concerning ``Factorisatio Numerorum'',
{\it J. Number Theory} {\bf 17} (1983), 1--28.

\bibitem{CH}
G. L. Cohen and P. Hagis Jr., On the number of prime factors of $n$ if $\vph(n)\mid (n-1)$,
{\it Nieuw Arch. Wisk.} (3) {\bf 28} (1980), 177--185.

\bibitem{EGPS}
P. Erd\H{o}s, A. Granville, C. Pomerance, and C. Spiro,
On the normal behavior of the iterates of some arithmetic functions,
in Bruce C. Berndt, Harold G. Diamond, Heini Halberstam, and Adolf Hildebrand, eds.,
{\it Analytic Number Theory, Proceedings of a Conference in Honor of Paul T. Bateman}, {\it Progr. Math.}, Vol.~85,
Birkh\"{a}user, 1990, pp.~165--204.

\bibitem{GOM}
Jos\'{e} Mar\'{i}a Grau and Antonio M. Oller-Marc\'{e}n,
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{\it Integers} {\bf 12} (2012), {\#}A37.

\bibitem{HW}
G. H. Hardy and E. M. Wright, {\it An Introduction to the Theory of
Numbers}, 6th edition, Oxford University Press, 2008.

\bibitem{Leh}
D. H. Lehmer, On Euler's totient function,
{\it Bull. Amer. Math. Soc.} {\bf 38} (1932), 745--751.

\bibitem{LP}
Florian Luca and Carl Pomerance,
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{\it Bol. Soc. Mat. Mexicana} (3) {\bf 17} (2011), 13--21.

\bibitem{McN}
Nathan McNew, Radically weakening the Lehmer and Carmichael conditions,
{\it Int. J. Number Theory} {\bf 9} (2013), 1215--1224.

\bibitem{MNW}
Nathan McNew and Thomas Wright, Infinitude of $k$-Lehmer numbers which are
not Carmichael, {\it Int. J. Number Theory} {\bf 12} (2016), 1863--1869.

\bibitem{Opp}
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{\bf 2} (1927), 123--130.

\bibitem{Pin}
Richard G. E. Pinch, Mathematics research page,\\
\url{http://www.chalcedon.demon.co.uk/rgep/rcam.html}.

\bibitem{Pom}
Carl Pomerance, On composites $n$ for which $\vph(n)\mid (n-1)$, II,
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\bibitem{Ren}
John Renze, Computational evidence for Lehmer's totient conjecture,\\
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\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2020 {\it Mathematics Subject Classification}:
Primary 11A25; Secondary 11A05, 11N25.

\noindent \emph{Keywords: } Lehmer's problem, Euler's totient function, multiplicative partition.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A001055},
\seqnum{A020555},
\seqnum{A033833},
\seqnum{A173703},
\seqnum{A337316}, and
\seqnum{A338998}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received November 24 2020;
revised versions received  November 25 2020; February 25 2021.
Published in {\it Journal of Integer Sequences}, April 25 2021.

\bigskip
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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{https://cs.uwaterloo.ca/journals/JIS/}.
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