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\begin{center}
\vskip 1cm{\LARGE\bf Generalizations of Amicable Numbers}
\vskip 1cm
\large
Stoyan Dimitrov\\
Faculty of Applied Mathematics and Informatics\\
Technical University of Sofia \\
Blvd. St. Kliment Ohridski 8\\
Sofia 1000\\ 
Bulgaria\\
\href{mailto:email}{\tt sdimitrov@tu-sofia.bg} \\
and\\
Institute of Biophysics and Biomedical Engineering\\
Bulgarian Academy of Sciences\\
Acad. G. Bonchev Str. Bl. 105\\
Sofia 1113\\ 
Bulgaria \\
\href{mailto:email}{\tt xyzstoyan@gmail.com} \\ 
\end{center}

\vskip .2 in
\begin{abstract}
In this paper we propose new generalizations of amicable numbers. We also give examples and prove properties of these new concepts.
\end{abstract}

\section{Notation}

One of the most remarkable sums in number theory is the aliquot sum $s(n)$ of a positive integer $n$. 
It is the sum of all proper divisors of $n$, that is, all divisors of $n$ other than $n$ itself. Thus 
\begin{equation*}
s(n)=\sum _{{d|n,} \atop {d<n}}d.
\end{equation*}
As usual $\sigma(n)$ is the sum of all the divisors of $n$. Thus $\sigma(n)=s(n)+n$. 
We let $\zeta(s)$ denote the Riemann zeta function. Recall that
\begin{equation*}
\zeta(s)=\sum\limits_{n=1}^\infty\frac{1}{n^s}
\end{equation*}
for $\textmd{Re}(s)>1$. The abbreviations $\textmd{gcd}$ and $\textmd{lcm}$ stand for the greatest common divisor and the least common multiple.


\section{Introduction}

Two natural numbers $m$ and $n$ are said to be {\it amicable\/} if
\begin{equation*}
\sigma(m)=\sigma(n)=m+n.
\end{equation*}
If $m=n$, then $m$ is perfect. It is well known that the smallest perfect number is 6.
The smallest pair of amicable numbers, (220, 284), was known to the Pythagoreans.
Although more than 2000 years have passed, it is still unknown whether infinitely many amicable pairs exist.
In 1913 Dickson \cite{Dickson} defined the natural numbers $n_1,n_2, n_3$ to 
form an amicable triple if
\begin{equation*}
\sigma(n_1)=\sigma(n_2)=\sigma(n_3)=n_1+n_2++n_3
\end{equation*}
or equivalently 
\begin{equation*}
\left|\begin{array}{ccc}
s(n_1)=n_2+n_3\\
s(n_2)=n_1+n_3\\
s(n_3)=n_1+n_2
\end{array}\right..
\end{equation*}
The smallest amicable triple, in the sense defined by Dickson, is (1980, 2016, 2556). The same generalization to the $k$-tuples was also proposed by Dickson. 
Afterwards, Yanney \cite{Yanney}, using the same symbolism and terminology, defined the numbers $n_1,n_2, n_3$ to form an amicable triple if
\begin{equation*}
2\sigma(n_1)=2\sigma(n_2)=2\sigma(n_3)=n_1+n_2++n_3
\end{equation*}
or equivalently 
\begin{equation*}
\left|\begin{array}{ccc}
n_1=s(n_2)+s(n_3)\\
n_2=s(n_1)+s(n_3)\\
n_3=s(n_1)+s(n_2)
\end{array}\right..
\end{equation*}
The smallest amicable triple according to Yanney's definition is (238, 255, 371). The same generalization to the $k$-tuples was also proposed by Yanney. 
Carmichael \cite{Carmichael} defined a {\it multiply amicable pair\/} as two numbers $m$ and $n$ such that
\begin{equation*}
\sigma(m)=\sigma(n)=t(m+n),
\end{equation*}
where $t$ is a positive integer. The same generalization to the $k$-tuples was proposed by Mason \cite{Mason}. 
In 1995 Cohen, Gretton and Hagis \cite{Cohen} defined natural numbers $m$ and $n$ to be multiamicable if the sum of the
proper divisors of each is a multiple of the other. More precisely, the pair $(m, n)$ is said to be {\it $(\alpha, \beta)$-amicable\/} if
\begin{equation*}
\sigma(m)-m=\alpha n \quad \mbox{ and }   \quad \sigma(n)-n=\beta m.
\end{equation*}
Another interesting generalization is due to Bishop, Bozarth, Kuss and Peet \cite{Bozarth}. 
They defined a {\it feebly amicable $k$-tuple\/} as numbers $n_1,\ldots, n_k$ such that
\begin{equation*}
\frac{n_1}{\sigma(n_1)}+\cdots+\frac{n_k}{\sigma(n_k)}=1.
\end{equation*}
Clearly every amicable $k$-tuple satisfies the equation above. For $k=2$, the smallest feebly amicable pair is (4, 12).
In Section \ref{WHM}, we will generalize the definition given in \cite{Bozarth}.
We say that an integer $n$ is an amicable number if it belongs to an amicable pair, or equivalently 
\begin{equation*}
\sigma(\sigma(n)-n)=\sigma(n).
\end{equation*}
Let $A(x)$ denote the number of amicable numbers up to $x$.
In 1955, Erd\H{o}s \cite{Erdos1} proved that the asymptotic density of amicable integers relative to the positive integers is 0. 
That is, the ratio of the number of amicable integers less than $x$ to $x$ tends to zero as $x$ tends to infinity.
Subsequently, several papers \cite{Erdos2, Pomerance1, Pomerance2, Rieger} established progressively better upper bounds for $A(x)$ 
and the best result to date is due to Pomerance \cite{Pomerance3} with
\begin{equation*}
A(x)\leq\frac{x}{e^{\sqrt{\log x}}}.
\end{equation*}
Many articles are devoted to generalizations of amicable numbers.  
We point out the papers \cite{Beck, Garcia, Hagis1, Hagis2, Lal, Pollack}, but many other similar results can be found in the literature.
Motivated by the aforementioned investigations, in this paper we propose an extension of the notion of amicable numbers.


\section{Lemmas}

\begin{lemma}\label{Erdos}
For a fixed integer $N>0$, the set of integers $l$ for which $N$ does not divide $\sigma(l)$ has density $0$.
\end{lemma}
\begin{proof}
\cite[Lemma 2]{Erdos1} 
\end{proof}

\begin{lemma}\label{sigma1}
For each number $x\geq 1$, we have
\begin{equation*}
\sum\limits_{n\leq x}\frac{\sigma(n)}{n}<\zeta(2)x.
\end{equation*}
\begin{proof}
Using the well-known identity
\begin{equation}\label{identity}
\frac{\sigma(n)}{n}=\sum_{u|n}\frac{1}{u}
\end{equation}
we write
\begin{align*}
&\sum\limits_{n\leq x}\frac{\sigma(n)}{n}=\sum\limits_{n\leq x}\sum_{u|n}\frac{1}{u}=\sum\limits_{u\leq x}\frac{1}{u}\sum_{n\leq x\atop{u|n}}1
=\sum\limits_{u\leq x}\frac{1}{u}\bigg\lfloor\frac{x}{u}\bigg\rfloor\leq x\sum\limits_{u\leq x}\frac{1}{u^2}<\zeta(2)x.\\
\end{align*}
\end{proof}
\end{lemma}

\begin{lemma}\label{sigmak}
Let $k\geq2$ be integer. For each number $x\geq 1$, we have
\begin{equation*}
\sum\limits_{n\leq x}\frac{\sigma^k(n)}{n^k}<(\zeta(2))^k\zeta(2k-1)x.
\end{equation*}
\begin{proof}
Using \eqref{identity} and the well-known identity
\begin{equation*}
\textmd{lcm}(n_1,\ldots, n_k)\,\textmd{gcd}\bigg(\frac{m}{n_1},\ldots, \frac{m}{n_k}\bigg)=m, 
\end{equation*}
where
\begin{equation*}
m=n_1\cdots n_k
\end{equation*}
we deduce
\begin{align*}
\sum\limits_{n\leq x}\frac{\sigma^k(n)}{n^k}&=\sum\limits_{n\leq x}\sum_{n_1|n}\cdots\sum_{n_k|n}\frac{1}{n_1\cdots n_k}
=\sum_{n_1\leq x}\cdots\sum_{n_k\leq x}\frac{1}{n_1\cdots n_k}\sum_{n\leq x\atop{\textmd{lcm}(n_1, \ldots, n_k})|n}1  \\
&=\sum_{n_1\leq x}\cdots\sum_{n_k\leq x}\frac{1}{n_1\cdots n_k}\bigg\lfloor\frac{x}{\textmd{lcm}(n_1, \ldots, n_k)}\bigg\rfloor\\
&\leq x\sum_{n_1\leq x}\cdots\sum_{n_k\leq x}\frac{1}{n_1\cdots n_k\,\textmd{lcm}(n_1, \ldots, n_k)}\\
&=x\sum_{n_1\leq x}\cdots\sum_{n_k\leq x}\frac{\textmd{gcd}\left(\frac{m}{n_1},\ldots, \frac{m}{n_k}\right)}{n^2_1\cdots n^2_k}
<x\sum\limits_{n_1=1}^\infty\cdots\sum\limits_{n_k=1}^\infty\frac{\textmd{gcd}\left(\frac{m}{n_1},\ldots, \frac{m}{n_k}\right)}{n^2_1\cdots n^2_k}\\
&=x\sum\limits_{d=1}^\infty\sum\limits_{\textmd{gcd}\left(\frac{m}{n_1},\ldots, \frac{m}{n_k}\right)=d}\frac{d}{n^2_1\cdots n^2_k}
<x\sum\limits_{d=1}^\infty\sum\limits_{n_1=1}^\infty\cdots\sum\limits_{n_k=1}^\infty\frac{d}{(dn'_1)^2\cdots(dn'_k)^2}\\
&=x\sum\limits_{d=1}^\infty\frac{1}{d^{2k-1}}\sum\limits_{n_1=1}^\infty\cdots\sum\limits_{n_k=1}^\infty\frac{1}{(n'_1)^2\cdots(n'_k)^2}=(\zeta(2))^k\zeta(2k-1)x.
\end{align*}
\end{proof}
\end{lemma}

\section{$(\alpha_1,\ldots,\alpha_k)$-multiamicable $k$-tuples}  

We say that the natural numbers $n_1,\ldots, n_k$ form an 
{\it $(\alpha_1,\ldots,\alpha_k)$-multiamicable $k$-tuple\/} if
\begin{equation}\label{sigma1k}
\sigma(n_1)=\sigma(n_2)=\cdots=\sigma(n_k)=\alpha_1n_1+\alpha_2n_2+\cdots+\alpha_kn_k.
\end{equation}
Of course, $\alpha_1,\ldots,\alpha_k$ must be positive integers. 
If $k=1$ and $\alpha_1=2$ then $n_1$ is a perfect number. 
When $\alpha_1=\alpha_2=\cdots=\alpha_k$ then $n_1,\ldots, n_k$ is a multiply amicable $k$-tuple according to Mason's definition. 
There is no extra interest in allowing the possibility that $n_1=n_2=\cdots =n_k$, so we shall assume always that $n_1<n_2<\cdots<n_k$
with the $k$-tuple $(\alpha_1,\ldots,\alpha_k)$ ordered accordingly. Hence
\begin{equation}\label{abundant}
(\alpha_1+\cdots+\alpha_k)n_1<\sigma(n_j)<(\alpha_1+\cdots+\alpha_k)n_k
\end{equation}
for each $j\in[1, k]$, which means that $n_1$ is $(\alpha_1+\cdots+\alpha_k)$-abundant. 
Following the method in \cite{Dickson} and \cite{Mason} we shell use the next proposition to find $(\alpha_1,\ldots,\alpha_k)$-multiamicable $k$-tuples.
\begin{theorem}
Suppose the natural numbers $N_1,\ldots, N_k$,  $\alpha_1,\ldots,\alpha_k$ and $a$ satisfy $(a, N_1)=\cdots=(a, N_k)= 1$ and
\begin{equation*}
\frac{\sigma(a)}{a}=\frac{\alpha_1N_1+\cdots+\alpha_kN_k}{\sigma(N_1)}=\cdots=\frac{\alpha_1N_1+\cdots+\alpha_kN_k}{\sigma(N_k)}.
\end{equation*}
Then $aN_1,\ldots, aN_k$ are an $(\alpha_1,\ldots,\alpha_k)$-multiamicable $k$-tuple.
\end{theorem}

\begin{proof}
This follows  directly from the multiplicativity of $\sigma$.
\end{proof}
For $k=2$ several of these pairs $(aN_1, aN_2)$  that are $(\alpha_1, \alpha_2)$-multiamicable pairs are listed in Table \ref{Table1}.
\begin{center}
\scalebox{0.9}{%
\begin{tabular}[t]{|p{1em}|p{1em}|p{3.6em}|p{3.8em}|p{3.2em}|p{3.2em}|p{16em}|}
\hline 
$\alpha_1$ & $\alpha_2$ & $N_1$ & $N_2$ & $\sigma(a)/a$ & $a$ &  \hspace{22mm} $(aN_1, aN_2)$  \\
\hline
1 & 2 & $2^313$ & $2^2 29$ & 8/5 & $3\cdot5$ & \hspace{20mm} (1560, 1740) \\
\hline
1 & 2 & $2^2 3^25\cdot41$ & $2^5 3^5$ & 1 & 1 & \hspace{20mm}  (7380, 7776) \\
\hline
1 & 2 & $2^3 41$ & $2^2 89$ & 104/63 & $3^2 7$ & \hspace{18mm}  (20664, 22428) \\
\hline
1 & 2 &  $17\cdot37$ & 683 & 35/12 & $2^5 3^3$  & \hspace{16mm} (543456, 590112) \\
\hline
1 & 2 & $17\cdot37$ & 683 & 35/12 & $2^3 3^2 13$ & \hspace{16mm} (588744, 639288) \\
\hline
1 & 2 & $7^211\cdot17$ & $107\cdot113$ & 65/24 & $2^3 3^2$ & \hspace{16mm} (659736, 870552) \\
\hline
1 & 2 & $13\cdot89$ & $1259$ & 35/12 & $2^5 3^3$ & \hspace{16mm} (999648, 1087776) \\
\hline
2 & 1 & $2^2 5\cdot107$ & $2^5 71$ & 13/9 & $3^2$  & \hspace{18mm} (19260, 20448) \\
\hline
2 & 1 & $2^3 3^5 11$ &  $2^5 3^2 79$ & 1 & 1 & \hspace{18mm} (21384, 22752) \\
\hline
2 & 1 & $2^2 3^3 29$  & $2^3 3\cdot139$  & 8/7 & 7 & \hspace{18mm} (21924, 23352)  \\
\hline
2 & 1 & $17\cdot37\cdot59$ & $179\cdot227$ & 403/144 &  $2^4 3^2$  & \hspace{14mm} (5343984, 5851152) \\
\hline
1 & 3 & $3^3 5^3$ & $3^2 5\cdot79$ & 9/4 & $2^5 7$ & \hspace{16mm} (756000, 796320) \\
\hline
3 & 1 & $11\cdot29$ & $17\cdot19$ & 32/9 & $2^2 3^3 5\cdot7$ & \hspace{14mm} (1205820, 1220940) \\
\hline
3 & 1 & $7\cdot13^2$ & $11^3$ & 10/3 & $2^3 3^3 5$ & \hspace{14mm} (1277640, 1437480) \\
\hline
3 & 1 & $3^2 19\cdot41$ & $3^5 29$ & 18/7 &  $2^3 5\cdot7$ & \hspace{14mm} (1963080, 1973160) \\
\hline
\end{tabular}
}
\captionof{table}{}\label{Table1}
\end{center}
The sequence \seqnum{A383239} in the OEIS \cite{Sloane} consists of the larger components of $(1, 2)$-multiamicable pairs.
The next result concerns the density of $(\alpha, \beta)$-multiamicable pairs. Our argument is a modification of the one used by Cohen, Gretton, and Hagis in \cite{Cohen}.
\begin{theorem}\label{Mox}
Let $M(x)$ denote the number of $(\alpha, \beta)$-multiamicable pairs $m$, $n$ with $m<n$ and $m\leq x$. Then $M(x)=o(x)$ as $x\rightarrow \infty$.
\end{theorem}

\begin{proof}
According to \eqref{sigma1k} we have   
\begin{equation}\label{sigmamn}
\sigma(m)=\sigma(n)=\alpha m+\beta n.
\end{equation}
Assume that $K$ is a large positive  integer.
Let $M_1(x)$ denote the number of $(\alpha, \beta)$-multiamicable pairs $m$, $n$ with $m\leq x$ and $\sigma(m)/m\geq K$.
From \eqref{abundant} we derive
\begin{equation}\label{sigmamab}
\frac{\sigma(m)}{m}>\alpha+\beta.
\end{equation}
Therefore the number of $n$'s that can correspond to a given value of $m$ is less than 
\begin{equation*}
\frac{1}{2}\bigg(\frac{\sigma(m)}{m}-1\bigg)\bigg(\frac{\sigma(m)}{m}-2\bigg).
\end{equation*}
Now Lemma \ref{sigma1} and Lemma \ref{sigmak} give us
\begin{align}\label{M1est}
M_1(x)&\leq\frac{1}{2}\sum\limits_{m\leq x\atop{\sigma(m)/m\geq K}}\bigg(\frac{\sigma(m)}{m}-1\bigg)\bigg(\frac{\sigma(m)}{m}-2\bigg)\nonumber\\
&\leq\frac{1}{2K}\sum\limits_{m\leq x}\frac{\sigma(m)}{m}\bigg(\frac{\sigma(m)}{m}-1\bigg)\bigg(\frac{\sigma(m)}{m}-2\bigg)\nonumber\\
&<\frac{1}{K}\sum\limits_{m\leq x}\frac{\sigma(m)}{m}+\frac{3}{2K}\sum\limits_{m\leq x}\frac{\sigma^2(m)}{m^2}+\frac{1}{2K}\sum\limits_{m\leq x}\frac{\sigma^3(m)}{m^3}\nonumber\\
&<\frac{1}{K}\zeta(2)x+\frac{3}{2K}(\zeta(2))^2\zeta(3)x+\frac{1}{2K}(\zeta(2))^3\zeta(5)x.
\end{align}
Let $M_2(x)$ denote the number of $(\alpha, \beta)$-multiamicable pairs $m$, $n$ with $m \leq x$ and such that $\sigma(m)/m<K$. By \eqref{sigmamab} we get
\begin{equation*}
\alpha+\beta<K.
\end{equation*}
Consequently the number of $n$'s that can correspond to a fixed value of $m$ is less than $\frac{1}{2}(K-1)(K-2)$. 
Using this consideration, \eqref{sigmamn} and Lemma \ref{Erdos} we obtain 
\begin{align}\label{M2est}
M_2(x)&\leq\frac{1}{2}(K-1)(K-2)\sum\limits_{m\leq x\atop{K^4\mid\sigma(m)}}1+\frac{1}{2}(K-1)(K-2)\sum\limits_{m\leq x\atop{K^4\nmid\sigma(m)}}1 \nonumber\\
&\leq\frac{1}{2}(K-1)(K-2)\sum\limits_{m\leq x\atop{\alpha m+\beta n\equiv 0\, ( \textmd{mod}\,K^4)}}1+o(x) \nonumber
\end{align}
\begin{align}
&\leq\frac{1}{2}(K-1)(K-2)\bigg(\frac{x}{K^3}+\mathcal{O}\big(K\big)\bigg)+o(x) \nonumber\\
&\leq\frac{x}{K}+o(x).
\end{align}
Clearly
\begin{equation}\label{M12}
M(x)=M_1(x)+M_2(x).
\end{equation}
Summarizing \eqref{M1est}, \eqref{M2est}, \eqref{M12} and bearing in mind that $K$ can be taken arbitrarily large we establish  
\begin{equation*}
M(x)=o(x) \quad \mbox{ as } \quad x\rightarrow \infty.
\end{equation*}
This completes the proof of Theorem \ref{Mox}.
\end{proof}
The following conjecture provides a natural conclusion to this section.
\begin{conjecture}
For any fixed positive integers $\alpha_1,\ldots,\alpha_k$, there exist  infinitely many $(\alpha_1,\ldots,\alpha_k)$-multiamicable $k$-tuples. 
\end{conjecture}


\section{$(\alpha, \beta)$-amicable pairs}

Let $\alpha$ and $\beta$ be positive integers. We say that the numbers $m$ and $n$ form an {\it $(\alpha, \beta)$-amicable pair\/} if
\begin{equation*}
\sigma(\alpha n)-\alpha n=m \quad \mbox{ and }   \quad \sigma(\beta m)-\beta m=n
\end{equation*}
or equivalently 
\begin{equation*}
m=s(\alpha n) \quad \mbox{ and }   \quad n=s(\beta m).
\end{equation*}
When $\alpha=\beta=1$ then $m$ and $n$ are amicable. 
Several $(\alpha, \beta)$-amicable pairs are listed in Table \ref{Table2}.
\begin{center}
\begin{tabular}[t]{|p{0.5em}|p{0.5em}|p{12em}|}
\hline 
$\alpha$ & $\beta$ &\hspace{3mm} $(\alpha, \beta)$-amicable pairs \\
\hline
1 & 2 &  $(26, 46)$, $(296, 586)$ \\
\hline
1 & 3 &  $(3, 4)$, $(15, 33)$, $(5919, 7905)$ \\
\hline
\end{tabular}
\captionof{table}{}\label{Table2}
\end{center}
The sequence \seqnum{A384411} in the OEIS \cite{Sloane} consists of $(1, 2)$-amicable pairs.
The OEIS \cite{Sloane} sequence \seqnum{A383483} lists the smaller elements of $(1, 3)$-amicable pairs.

\section{$\PM(p, q)$-amicable $k$-tuples}

The abbreviation $\PM$ stands for power mean. Let $k\geq1$, $p\geq1$  and $q\geq1$ be integers.
We say that the numbers $n_1,\ldots, n_k$ form a
{\it $\PM(p, q)$--amicable $k$-tuple} if
\begin{equation*}
\sigma^p(n_1)+\sigma^p(n_2)+\cdots+\sigma^p(n_k)=q(n_1+n_2+\cdots+n_k)^p.
\end{equation*}
Obviously, when $k=q$, then every amicable  $k$-tuple satisfies the above equation.
Several $\PM(p, q)$-amicable $k$-tuples that are not amicable  $k$-tuples are listed in Table \ref{Table3}.

\begin{center}
\begin{tabular}[t]{|p{0.5em}|p{0.5em}|p{0.5em}|p{22em}|}
\hline 
$k$ & $p$ & $q$ & \hspace{18mm} $\PM(p, q)$-amicable $k$-tuples  \\
\hline
2 & 1 & 2 & $(3, 20)$, $(5, 12)$, $(5, 70)$, $(5, 88)$, $(6, 28)$, $(10, 20)$ \\
\hline
2 & 1 & 3 & $(6, 180)$, $(10, 780)$, $(24, 780)$, $(26, 660)$, $(34, 504)$ \\
\hline
2 & 2 & 1 & $(2, 3)$, $(19, 33)$, $(27, 77)$, $(39, 161)$, $(45, 133)$, $(51, 69)$ \\
\hline
2 & 2 & 2 & $(1, 4)$, $(1378, 9962)$,   $(1660, 4892)$, $(1975, 10425)$   \\
\hline
3 & 1 & 2 &  $(1, 2, 20)$, $(1, 3, 18)$, $(1, 4, 20)$,  $(1, 8, 20)$, $(1, 10, 18 )$\\
\hline
3 & 1 & 3 &  $(1, 76, 360)$, $(2, 11, 240)$, $(2, 41, 420)$,  $(6, 120, 180)$ \\
\hline
3 & 2 & 1 &  $(1, 47, 185)$, $(2, 11, 14)$, $(2, 110, 371)$, $(3, 302, 411)$  \\
\hline
3 & 2 & 2 &  $(3, 36, 98)$, $(5, 34, 135)$, $(5, 40, 105)$, $(10, 106, 406)$  \\
\hline
3 & 2 & 3 &  $(14, 350, 1340)$, $(22, 96, 1862)$, $(31, 301, 1876)$   \\
\hline
3 & 2 & 4 &  $(3, 39, 156)$, $(11, 40, 294)$, $(12, 14, 60)$, $(17, 70, 210)$  \\
\hline
3 & 2 & 5 & $(6, 222, 1608)$, $(15, 33, 168)$, $(30, 66, 552)$  \\
\hline
3 & 3 & 1 &  $(2, 10, 15)$, $(4, 20, 39)$, $(8, 40, 87)$, $(9, 45, 63)$ \\
\hline
3 & 3 & 3 & $(56, 134, 710)$, $(108, 268, 1724)$, $(236, 404, 2510)$  \\
\hline
\end{tabular}
\captionof{table}{}\label{Table3}
\end{center}
The sequence \seqnum{A383484} in the OEIS \cite{Sloane} consists of the larger components of $\PM(2, 1)$-amicable pairs.
The OEIS \cite{Sloane} sequence \seqnum{A385008} lists the larger elements of $\PM(2, 2)$-amicable pairs.

\section{$\WPM(p)$-amicable $k$-tuples}

The abbreviation $\WPM$ stands for weighted power mean. Let $k\geq1$ and $p\geq1$ be integers. 
We say that the numbers $n_1,\ldots, n_k$ form an 
{\it $\WPM(p)$-amicable $k$-tuple\/} if
\begin{equation*}
n_1\sigma^p(n_1)+n_2\sigma^p(n_2)+\cdots+n_k\sigma^p(n_k)=(n_1+n_2+\cdots+n_k)^{p+1}.
\end{equation*}
Clearly each amicable $k$-tuple satisfies the above equation.
Several $\WPM(p)$-amicable $k$-tuples that are not amicable $k$-tuples are listed in Table \ref{Table4}.

\begin{center}
\begin{tabular}[t]{|p{0.5em}|p{0.5em}|p{19em}|}
\hline 
$k$ & $p$ &\hspace{10mm} $\WPM(p)$-amicable $k$-tuples  \\
\hline
2 & 1 &  $(4, 6)$, $(10, 16)$, $(34, 68)$, $(60, 81)$, $(91, 273)$  \\
\hline
2 & 2 &  $(7, 21)$, $(105, 231)$, $(1065, 2499)$ \\
\hline
3 & 1 &  $(1, 21, 63)$, $(1, 22, 44)$, $(2, 38, 98)$, $(4, 6, 34)$ \\
\hline
3 & 2 & $(12, 276, 412)$, $(70, 210, 224)$, $(87, 189, 264)$  \\
\hline
\end{tabular}
\captionof{table}{}\label{Table4}
\end{center}
The sequence \seqnum{A383714} in the OEIS \cite{Sloane} consists of the larger components of $\WPM(2)$-amicable pairs.


\section{$\GM$-amicable $k$-tuples}

The abbreviation $\GM$ stands for geometric mean. Let $k\geq1$ be integer. We say that the numbers $n_1,\ldots, n_k$ form a {\it $\GM$-amicable $k$-tuple\/} if
\begin{equation*}
\sigma(n_1)\sigma(n_2)\cdots\sigma(n_k)=(n_1+n_2+\cdots+n_k)^k.
\end{equation*}
It is easy to see that every amicable  $k$-tuple satisfies the above equation.
Several $\GM$-amicable $k$-tuples that are not amicable $k$-tuples are listed in Table \ref{Table5}.

\begin{center}
\begin{tabular}[t]{|p{0.5em}|p{27.2em}|}
\hline 
$k$ & \hspace{33mm}  $\GM$-amicable $k$-tuples  \\
\hline
2 & $(28, 84)$, $(42, 102)$, $(60, 276)$, $(92, 160)$, $(244, 624)$, $(426, 582)$ \\
\hline
3 & $(1080, 1092, 1188)$, $(10164, 10584, 11172)$, $(10440, 10692, 11628)$ \\
\hline
\end{tabular}
\captionof{table}{}\label{Table5}
\end{center}
The OEIS \cite{Sloane} sequences \seqnum{A383932} and \seqnum{A386010} list the larger elements of $\GM$-amicable pairs and $\GM$-amicable triples, respectively.


\section{$\WGM$-amicable $k$-tuples}

The abbreviation $\WGM$ stands for weighted geometric mean. Let $k\geq1$ be integer. We say that the numbers $n_1,\ldots, n_k$ form a
{\it $\WGM$-amicable $k$-tuple\/} if
\begin{equation*}
\sigma(n_1)^{n_1}\sigma(n_2)^{n_2}\cdots\sigma(n_k)^{n_k}=(n_1+n_2+\cdots+n_k)^{n_1+n_2+\cdots+n_k}.
\end{equation*}
It is clear that each amicable  $k$-tuple satisfies the above equation.
Although we cannot set an example, we cannot rule out the existence of $\WGM$-amicable $k$-tuples that are not amicable $k$-tuples.
The sequence \seqnum{A385186} in the OEIS \cite{Sloane} consists of the larger components of $\WGM$-amicable pairs.


\section{$\IWGM$-amicable $k$-tuples}

The abbreviation $\IWGM$ stands for inverse weighted geometric mean. The positive integers $n_1,\ldots, n_k$ form an 
{\it $\IWGM$-amicable $k$-tuple} if
\begin{equation*}
n_1^{\sigma(n_1)}n_2^{\sigma(n_2)}\cdots n_k^{\sigma(n_k)}=(n_1n_2\cdots n_k)^{n_1+n_2+\cdots+n_k}.
\end{equation*}
Clearly each amicable  $k$-tuple satisfies the above equation. Several $\IWGM$-amicable pairs that are not amicable pairs are listed in Table \ref{Table6}.
\begin{center}
\begin{tabular}[t]{|p{37em}|}
\hline 
\hspace{54mm} $\IWGM$-amicable pairs \\
\hline
 $(1, 2)$, $(1, 3)$, $(1, 5)$, $(1, 7)$, $(1, 11)$, $(1, 13)$, $(1, 17)$, $(1, 19)$, $(1, 23)$, $(1, 29)$, $(1, 31)$, $(1, 37)$  \\     
\hline
\end{tabular}
\captionof{table}{}\label{Table6}
\end{center}
The OEIS \cite{Sloane} sequence \seqnum{A385492} lists the larger elements of $\IWGM$-amicable pairs.


\section{$\HM(p, q)$-amicable $k$-tuples}

The abbreviation $\HM$ stands for harmonic mean. Let $k\geq1$, $p\geq1$  and $q\geq1$ be integers. We say that the numbers $n_1,\ldots, n_k$ form a 
{\it $\HM(p, q)$-amicable $k$-tuple} if
\begin{equation*}
\Bigg(\frac{1}{\sigma^p(n_1)}+\frac{1}{\sigma^p(n_2)}+\cdots+\frac{1}{\sigma^p(n_k)}\Bigg)(n_1+n_2+\cdots+n_k)^p=q.
\end{equation*}
Obviously, if $k=q$, then every amicable  $k$-tuple satisfies the above equation.
Several $\HM(p, q)$-amicable $k$-tuples that are not amicable  $k$-tuples are listed in Table \ref{Table7}.

\begin{center}
\begin{tabular}[t]{|p{0.5em}|p{0.5em}|p{0.5em}|p{24em}|}
\hline 
$k$ & $p$ & $q$ & \hspace{22mm} $\HM(p, q)$-amicable $k$-tuples  \\
\hline
2 & 1 & 2 & $(20, 28)$, $(24, 56)$, $(30, 66)$, $(40, 90)$, $(56, 88)$, $(92, 132)$ \\
\hline
2 & 1 & 3 & $(3, 6)$, $(10, 32)$, $(12, 60)$, $(15, 33)$, $(24, 116)$, $(33, 57)$ \\
\hline
2 & 2 & 1 & $(120, 168)$, $(1272, 1320)$, $(2160, 3792)$, $(3672, 4968)$ \\
\hline
2 & 2 & 4 & $(435, 717)$, $(447, 513)$, $(2001, 2607)$, $(2001, 2607)$ \\
\hline
3 & 1 & 3 &  $(840, 1020, 1380)$, $(1008, 1260, 1638)$, $(2016, 2232, 2772)$\\
\hline
3 & 2 & 4 &  $(480, 480, 1056)$, $(1400, 1400, 2160)$, $(3936, 3936, 6240)$\\
\hline
\end{tabular}
\captionof{table}{}\label{Table7}
\end{center}
The sequence \seqnum{A384814} in the OEIS \cite{Sloane} consists of the larger components of $\HM(1, 2)$-amicable pairs.
The OEIS \cite{Sloane} sequence \seqnum{A383964} lists the larger elements of $\HM(2, 1)$-amicable pairs.
The sequence \seqnum{A385155} in the OEIS \cite{Sloane} consists of the larger members of $\HM(1, 3)$-amicable triples.

\section{$\WHM(p)$-amicable $k$-tuples}\label{WHM}

The abbreviation $\WHM$ stands for weighted harmonic mean. 
Let $k\geq1$ and $p\geq1$ be integers. We say that the numbers $n_1,\ldots, n_k$ form a {\it $\WHM(p)$-amicable $k$-tuple\/} if
\begin{equation*}
\Bigg(\frac{n^p_1}{\sigma^p(n_1)}+\frac{n^p_2}{\sigma^p(n_2)}+\cdots+\frac{n^p_k}{\sigma^p(n_k)}\Bigg)(n_1+n_2+\cdots+n_k)^p=n^p_1+n^p_2+\cdots+n^p_k.
\end{equation*}
Clearly each amicable $k$-tuple satisfies the above equation.
It is easy to see that $\WHM(1)$-amicable $k$-tuple coincides with feebly amicable $k$-tuple defined in \cite{Bozarth}.
Several $\WHM(p)$-amicable $k$-tuples that are not amicable $k$-tuples are listed in Table \ref{Table8}.

\begin{center}
\begin{tabular}[t]{|p{0.5em}|p{0.5em}|p{25em}|}
\hline 
$k$ & $p$ &\hspace{24mm} $\WHM(p)$-amicable $k$-tuples \\
\hline
3 & 1 & $(72, 360, 504)$, $(84, 120, 840)$, $(84, 672, 840)$, $(96, 660, 840)$   \\    
\hline
3 & 2 & $(117, 117, 4680)$\\    
\hline
\end{tabular}
\captionof{table}{}\label{Table8}
\end{center}
The OEIS \cite{Sloane} sequences \seqnum{A384487} and \seqnum{A385749} list the larger elements of $\WHM(1)$- and  $\WHM(2)$-amicable triples, respectively.


\section{Cross-harmonious pairs}

Let $a$ and $b$ be positive integers. We say that $a$ and $b$ form a 
{\it cross-harmonious pair\/} if 
\begin{equation*}
\frac{b}{\sigma(a)}+\frac{a}{\sigma(b)}=1.
\end{equation*}
Clearly, every amicable pair is cross-harmonious. Several cross-harmonious pairs that are not amicable pairs are listed in Table \ref{Table9}.
\begin{center}
\begin{tabular}[t]{|p{35em}|}
\hline 
\hspace{52mm} Cross-harmonious pairs \\
\hline
 $(12, 14)$, $(12, 20)$, $(30, 54)$, $(32, 42)$, $(48, 62)$, $(70, 88)$, $(70, 108)$, $(78, 126)$, $(88, 114)$  \\     
\hline
\end{tabular}
\captionof{table}{}\label{Table9}
\end{center}
The sequence \seqnum{A384706} in the OEIS \cite{Sloane} consists of the larger components of cross-harmonious pairs.


\section{$\MP(p, q)$-amicable $k$-tuples}

The abbreviation $\MP$ stands for mean power. Let $k\geq1$, $p\geq2$  and $q\geq1$ be integers. We say that the numbers $n_1,\ldots, n_k$ form an
{\it $\MP(p, q)$-amicable $k$-tuple\/} if
\begin{equation*}
\sigma^p(n_1)+\sigma^p(n_2)+\cdots+\sigma^p(n_k)=q(n^p_1+n^p_2+\cdots+n^p_k).
\end{equation*}
Several $\MP(p, q)$-amicable $k$-tuples are listed in Table \ref{Table10}.

\begin{center}
\begin{tabular}[t]{|p{0.5em}|p{0.5em}|p{0.5em}|p{30.6em}|}
\hline 
$k$ & $p$ & $q$ & \hspace{34mm} $\MP(p, q)$-amicable $k$-tuples  \\
\hline
2 & 2 & 2 & $(1, 2)$, $(13, 21)$, $(13, 27)$, $(17, 175)$, $(45, 123)$, $(1069, 2133)$, $(1093, 2187)$ \\
\hline
2 & 2 & 4 & $ (6, 28)$, $(12, 14)$, $(48, 62)$, $(112, 124)$, $(135, 208)$, $(160, 189)$, $(192, 254)$ \\
\hline
3 & 2 & 2 & $(2, 4, 51)$, $(3, 40, 71)$, $(5, 12, 23)$, $(7, 116, 303)$, $(11, 20, 55)$, $(12, 215, 333)$\\
\hline
3 & 2 & 3 & $(1, 81, 148)$, $(10, 94, 164)$, $(14, 20, 34)$, $(20, 82, 116)$, $(26, 70, 418)$ \\
\hline
4 & 3 & 3 & $(1, 4, 5, 9)$, $(32, 34, 202, 245)$, $(52, 108, 135, 233)$, $(55, 58, 230, 281)$ \\
\hline
4 & 3 & 4 & $(2, 49, 56, 118)$, $(7, 35, 51, 75)$, $ (10, 168, 207, 307)$, $(37, 74, 232, 253)$\\
\hline
4 & 3 & 5 & $(11, 25, 95, 148)$, $(15, 59, 128, 129)$, $(35, 170, 186, 237)$, $(44, 125, 139, 266)$ \\
\hline
4 & 3 & 6 & $(6, 80, 85, 135)$, $(20, 22, 34, 92)$, $(41, 66, 123, 190)$, $(41, 70, 107, 190)$ \\
\hline
4 & 3 & 7 & $(3, 181, 198, 212)$, $(11, 21, 24, 25)$, $(14, 30, 94, 102)$, $(23, 81, 112, 135)$ \\
\hline
4 & 3 & 8 & $(13, 91, 116, 176)$, $(30, 74, 82, 112)$, $(48, 62, 112, 124)$, $(48, 62, 160, 189)$ \\
\hline
4 & 3 & 9 & $(18, 55, 178, 220)$, $(28, 231, 273, 320)$, $(36, 43, 44, 78)$, $(44, 46, 96, 258)$ \\
\hline
\end{tabular}
\captionof{table}{}\label{Table10}
\end{center}
The OEIS \cite{Sloane} sequence \seqnum{A384255} lists the larger members of $\MP(2,2)$-amicable pairs.
\begin{question}
Is there an $\MP(2, 2)$-amicable pair $(m, n)$ such that $\sigma(m)=\sigma(n)$?
More precisely, we raise the question of the existence of a number pair $(m, n)$ such that
\begin{equation*}
\sigma^2(m)=\sigma^2(n)=m^2+n^2.
\end{equation*}
\end{question}


\section{Sigma-quadratic triples}

The positive integers $a\leq b$ and $c$ form a 
{\it sigma-quadratic triple\/} if 
\begin{equation*}
\sigma^2(a)=\sigma^2(b)=a^2+b^2+c^2.
\end{equation*}
Several sigma-quadratic triples are listed in Table \ref{Table11}.
\begin{center}
\begin{tabular}[t]{|p{37em}|}
\hline 
\hspace{52mm} Sigma-Quadratic triples \\
\hline
$(2, 2, 1)$, $(40, 40, 70)$, $(40, 58, 56)$, $(164, 194, 148)$, $(196, 196, 287)$, $(224, 224, 392)$, $(1120, 1120, 2576)$, 
$(3040, 4024, 5632)$, $(44200, 49300, 96680)$, $(687184, 703312, 965780)$ \\     
\hline
\end{tabular}
\captionof{table}{}\label{Table11}
\end{center}
The sequence \seqnum{A385356} in the OEIS \cite{Sloane} consists of the first components of sigma-quadratic triples.



\section{Sigma-quadratic quadruples}

The positive integers $a\leq b\leq c$ and $d$ form a 
{\it sigma-quadratic quadruple\/} if 
\begin{equation*}
\sigma^2(a)=\sigma^2(b)=\sigma^2(c)=a^2+b^2+c^2+d^2.
\end{equation*}
Several sigma-quadratic quadruples are listed in Table \ref{Table12}.
\begin{center}
\begin{tabular}[t]{|p{35em}|}
\hline 
\hspace{50mm} Sigma-Quadratic quadruples \\
\hline
$(4, 4, 4, 1)$, $(204, 220, 220, 340)$, $(3480, 3672, 4296, 8520)$, $(4796, 4988, 5276, 3108)$, $(5532, 6812, 6812, 6628)$, $(7152, 9272, 9584, 10816)$, $(33468, 35380, 43676, 42700)$\\     
\hline
\end{tabular}
\captionof{table}{}\label{Table12}
\end{center}

The OEIS \cite{Sloane} sequence \seqnum{A385531} lists the first elements of sigma-quadratic quadruples.

\section{Sigma-cubic triples}

The positive integers $a$, $b$, and $c$ form a {\it sigma-cubic triple\/} if 
\begin{equation*}
\sigma^3(a)=a^3+b^3+c^.
\end{equation*}
Several sigma-cubic triples are listed in Table \ref{Table13}.
\begin{center}
\begin{tabular}[t]{|p{32em}|}
\hline 
\hspace{50mm} Sigma-Cubic triples \\
\hline
$(5, 3, 4)$, $(6, 8, 10)$, $(53, 12, 19)$, $(58, 59, 69)$, $(102, 26, 208)$, $(102, 117, 195)$, $(118, 116, 138)$, $(152, 172, 264)$, $(168, 336, 408)$, $(197, 27, 46)$, $(214, 173, 267)$ \\     
\hline
\end{tabular}
\captionof{table}{}\label{Table13}
\end{center}
The sequence \seqnum{A385325} in the OEIS \cite{Sloane} consists of the first members of sigma-cubic triples.


\section{Sigma-cubic quadruples}

The positive integers $a\leq b$, $c$ and $d$ form a 
{\it sigma-cubic quadruple\/} if 
\begin{equation*}
\sigma^3(a)=\sigma^3(b)=a^3+b^3+c^3+d^3.
\end{equation*}
Several sigma-cubic quadruples are listed in Table \ref{Table14}.
\begin{center}
\begin{tabular}[t]{|p{27em}|}
\hline 
\hspace{35mm} Sigma-Cubic quadruples \\
\hline
$(153, 153, 105, 165)$, $(216, 216, 168, 576)$, $(255, 321, 84, 312)$, $(324, 324, 271, 804)$, $(672, 910, 147, 1925)$, $(735, 1243, 215, 615)$ \\     
\hline
\end{tabular}
\captionof{table}{}\label{Table14}
\end{center}
The OEIS \cite{Sloane} sequence \seqnum{A385397} lists the first components of sigma-cubic quadruples.

\section{Sigma-cubic quintuples}

The positive integers $a\leq b\leq c$ and $d\leq e$ 
form a {\it sigma-cubic quintuple\/} if 
\begin{equation*}
\sigma^3(a)=\sigma^3(b)=\sigma^3(c)=a^3+b^3+c^3+d^3+e^3.
\end{equation*}
Several sigma-cubic quintuples are listed in Table \ref{Table15}.
\begin{center}
\begin{tabular}[t]{|p{35em}|}
\hline 
\hspace{50mm} Sigma-Cubic quintuples \\
\hline
$(30, 55, 55, 11, 23)$, $(62, 62, 69, 4, 43)$, $(90, 153, 153, 135, 135)$, $(174, 190, 323, 5, 94)$, $(238, 321, 321, 77, 81)$, $(357, 385, 385, 233, 266)$, $(390, 476, 598, 470, 814)$\\     
\hline
\end{tabular}
\captionof{table}{}\label{Table15}
\end{center}
The OEIS \cite{Sloane} sequence \seqnum{A386378} consists of the first elements of sigma-cubic quintuples.


\section{$\Prime(i, j)$-amicable pairs}

The abbreviation $\Prime$ stands for prime. We say that the prime numbers $p$ and $q$, where $p<q$, form a {\it $\Prime(i, j)$-amicable pair\/} if
\begin{equation*}
\sigma(p+i)=\sigma(q+j)=p+q.
\end{equation*}
It is easy to see that when $i+j=0$ and $(p, q)$ as a $\Prime(i, j)$-amicable pair then $(p+i, q+j)$ is an amicable pair. 
Several  $\Prime(i, j)$-amicable pairs are listed in Table \ref{Table16}.
\begin{center}
\begin{tabular}[t]{|p{0.9em}|p{0.9em}|p{34.5em}|}
\hline 
$i$ & $j$ &\hspace{48mm} $\textmd{P(i, j)}$-amicable pairs \\
\hline
1 & 1 &  $(23, 37)$, $(34673, 34687)$, $(55373, 65587)$, $(2056961, 2089951)$, $(5174363, 8161477)$  \\    
\hline
-1 & 1 & $(5021, 5563)$, $(185369, 203431)$, $(308621, 389923)$, $(879713, 901423)$\\    
\hline
-1 & -1 & $(853, 1163)$, $(4513, 7583)$, $(9109, 17099)$, $(44917, 65963)$, $(46183, 48857)$\\    
\hline
1 & 2 & $(179, 367)$, $(263, 457)$, $(557, 691)$, $(332273, 341647)$, $(401309, 909091)$\\    
\hline
1 & 3 & $(19, 23)$, $(29, 43)$, $(137, 151)$, $(223, 281)$, $(727, 953)$, $(22651, 22709)$ \\    
\hline
3 & 1 & $(11, 13)$, $(41, 43)$, $(107, 109)$, $(149, 151)$, $(257, 331)$, $(881, 883)$, $(2141, 2143)$ \\    
\hline
3 & 3 & $(2377, 3671)$, $(7069, 8807)$, $(7949, 9907)$, $(44701, 52067)$, $(71761, 85487)$ \\    
\hline
\end{tabular}
\captionof{table}{}\label{Table16}
\end{center}
The sequences \seqnum{A385586}, \seqnum{A385718}, \seqnum{A385739} and \seqnum{A385740} in the OEIS \cite{Sloane} list the larger elements of 
$\Prime(1, 1)$-, $\Prime(1, 2)$-, $\Prime(-1, 1)$- and $\Prime(-1, -1)$-amicable pairs, respectively.


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\bibitem{Bozarth} J. Bishop, A. Bozarth, R. Kuss, and B. Peet, The abundancy index and feebly amicable numbers, 
{\it Ball State Undergraduate Mathematics Exchange} \textbf{15} (2021), 65--77.

\bibitem{Carmichael} R. D. Carmichael, History of the theory of numbers, 
{\it Amer. Math. Monthly} \textbf{26} (1919), 396--403.

\bibitem{Cohen} G. L. Cohen, S. Gretton, and P.  Hagis, Multiamicable numbers,
{\it Math. Comp.} \textbf{64} (1995), 1743--1753.

\bibitem{Dickson} L. E. Dickson, Amicable number triples,
{\it Amer. Math. Monthly} \textbf{20} (1913), 84--92.

\bibitem{Erdos1} P. Erd\H{o}s, On amicable numbers, 
{\it Publ. Math. Debrecen} \textbf{4} (1955), 108--111. 

\bibitem{Erdos2} P. Erd\H{o}s and G. Rieger, Ein Nachtrag \"uber befreundete Zahlen, 
{\it J. Reine Angew. Math.} \textbf{273} (1975), 220--220.

\bibitem{Garcia} M. Garcia, J. Pedersen, and H. Riele, Amicable pairs, a survey, 
{\it Fields Inst. Commun.} \textbf{41} (2004), 179--196.

\bibitem{Hagis1} P. Hagis, Unitary amicable numbers, 
{\it Math. Comp.} \textbf{25} (1971), 915--918.

\bibitem{Hagis2} P. Hagis and G. Lord, Quasi-amicable numbers, 
{\it Math. Comp.} \textbf{31}, (1977), 608--611.

\bibitem{Lal} M. Lal and A. Forbes,  A note on Chowla's function, 
{\it Math. Comp.} \textbf{25} (1971), 923--925.

\bibitem{Mason} T. E. Mason, On amicable numbers and their generalizations,
{\it Amer. Math. Monthly} \textbf{28} (1921), 195--200.

\bibitem{Pollack} P. Pollack, Quasi-amicable numbers are rare, 
{\it J. Integer Sequences} \textbf{14} (2011), Article 11.5.2.

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{\it J. Reine Angew. Math.} \textbf{293/294} (1977), 217--222. 

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{\it Analytic Number Theory} (Springer, Cham, 2015), 321--327.

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{\it J. Reine Angew. Math.} \textbf{261} (1973), 157--163. 

\bibitem{Sloane} N. J. A. Sloane et al., The On-Line Encyclopedia of Integer 
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\end{thebibliography}


\bigskip
\hrule
\bigskip


\noindent 2020 {\it Mathematics Subject Classification}:
Primary 11A25; Secondary 11D72.

\noindent \emph{Keywords:}  sum of divisors, amicable numbers.

\bigskip
\hrule
\bigskip


\noindent (Concerned with sequences
\seqnum{A383239},
\seqnum{A383483},
\seqnum{A383484},
\seqnum{A383714},
\seqnum{A383932},
\seqnum{A383964},
\seqnum{A384255},
\seqnum{A384411},
\seqnum{A384487},
\seqnum{A384706},
\seqnum{A384814},
\seqnum{A385008},
\seqnum{A385155},
\seqnum{A385186},
\seqnum{A385325},
\seqnum{A385356},
\seqnum{A385397},
\seqnum{A385492},
\seqnum{A385531},
\seqnum{A385586},
\seqnum{A385718},
\seqnum{A385739},
\seqnum{A385740},
\seqnum{A385749},
\seqnum{A386010}, and
\seqnum{A386378}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received August 15 2024;
revised versions received August 16 2024; July 28 2025; July 30 2025.
Published in {\it Journal of Integer Sequences},
August 1 2025.

\bigskip
\hrule
\bigskip


\noindent
Return to \href{https://cs.uwaterloo.ca/journals/JIS/}{Journal of Integer Sequences home page}.
\vskip .1in

                                                                                

\end{document} 



