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\begin{center}
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{\LARGE\bf Fixed Points of Augmented Generalized \\
\vskip .1in
Happy Functions II: Oases and Mirages 
}
\vskip 1cm

\begin{minipage}[t]{0.5\textwidth}
\begin{center}
Breeanne Baker Swart \\
Department of Mathematical Sciences \\
The Citadel\\
171 Moultrie St. \\
Charleston, SC 29409 \\
USA \\
\href{mailto:breeanne.swart@citadel.edu}{\tt breeanne.swart@citadel.edu} \\
\ \\
Susan Crook\\
Division of Mathematics, Engineering and Computer Science\\
Loras College\\ 
1450 Alta Vista St.\\
Dubuque, IA 52001\\
USA\\
\href{mailto:susan.crook@loras.edu}{\tt susan.crook@loras.edu} \\
\ \\
Helen G. Grundman\\
Department of Mathematics\\
Bryn Mawr College\\
101 N. Merion Ave.\\
Bryn Mawr, PA 19010\\
USA\\
\href{mailto:grundman@brynmawr.edu}{\tt grundman@brynmawr.edu} \\
\end{center}
\end{minipage}
\begin{minipage}[t]{0.45\textwidth}
\begin{center}
Laura Hall-Seelig\\
Department of Mathematics\\
Merrimack College\\
315 Turnpike Street\\
North Andover, MA 01845\\
USA\\
\href{mailto:hallseeligl@merrimack.edu}{\tt hallseeligl@merrimack.edu} \\

\ \\
May Mei\\ 
Department of Mathematics and Computer Science\\
Denison University\\
100 West College Street\\
Granville, Ohio 43023\\
USA\\
\href{mailto:meim@denison.edu}{\tt meim@denison.edu} \\
\ \\
Laurie Zack \\
Department of Mathematical Sciences\\ 
High Point University\\
One University Parkway\\
High Point, NC 27268\\
USA\\
\href{mailto:lzack@highpoint.edu}{\tt lzack@highpoint.edu} \\
\end{center}
\end{minipage}
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\vskip .2in


\begin{abstract}
An augmented generalized happy function $\Scb$ maps a positive integer to the sum of the squares of its base $b$ digits plus $c$. For $b\geq 2$ and $k \in \ZZ^+$, a \emph{$k$-desert base $b$} is a set of $k$ consecutive non-negative integers $c$ for each of which $\Scb$ has no fixed points. 
In this paper, we examine a complementary notion, a \emph{$k$-oasis base $b$}, which we define to be a set of $k$ consecutive non-negative integers $c$ for each of which $\Scb$ has a fixed point. In particular, after proving some basic properties of oases base $b$, we compute bounds on the lengths of oases base $b$ and compute the minimal examples of maximal length oases base $b$ for small values of $b$.  
\end{abstract}




\section{Introduction}\label{S:Introduction}


The concepts of happy number~\seqnum{A007770} and generalized happy number~\cite{genhappy,guy,honsberger} were generalized further~\cite{augment} by allowing for augmentation, as follows. 
\begin{definition}
For integers $c \geq 0$ and $b \geq 2$, the {\it augmented generalized happy function}, $\Scb:{\ZZ}^+ \rightarrow {\ZZ}^+$, is defined for $0\leq a_i \leq b-1$ and $a_n \neq 0$ by
\begin{equation*}\label{eq:definition}
\Scb\left(\sum_{i=0}^n b^i a_i  \right) =
c + \sum_{i=0}^n a_i^2.
\end{equation*}
The value $c$ is called the {\it augmenting constant} of $\Scb$.  A positive integer $a$ is called a {\it fixed point} of $\Scb$ if $\Scb(a) = a$.
A positive integer $a$ is a {\it happy number} if, for some $k\in \ZZ^+$, $S_{[0,10]}^k(a) = 1$.  
\end{definition}


The function $S_{[0,10]}$ is easily seen to have exactly one fixed point, while, depending on the values of $c$ and $b$, the function $\Scb$ may have zero, one, or multiple fixed points~\cite{desert}.  The case of zero fixed points is studied in Part I of this paper~\cite{desert}, in which Baker Swart et al.\ prove that for each $b \geq 2$, there exist arbitrarily long finite sequences of consecutive values of $c$ for which $\Scb$ has no fixed point.

In this work, we study the complementary case by considering sets of consecutive augmenting constants $c$ for which $\Scb$ has at least one fixed point and proving that, for each fixed $b$, the size of these sets is bounded.
In Section~\ref{S:OasisB}, we define the concept of $k$-oasis base $b$, determine some initial properties, and prove a bound, for each $b\geq 2$, on the lengths of oases base $b$.
In Section~\ref{S:MirageB}, we define the concept of a $k$-mirage base $b$, prove that the maximal length of mirages base $b$ bounds the maximal length of oases base $b$, and provide an algorithm for finding the maximal length of mirages base $b$.  Finally, we use the above to determine the maximal length of oases (and of mirages) base $b$, for all $b \leq 20$.

For later convenience, we note that if
$\sum_{i=0}^{n}b^ia_i$ is a fixed point of $\Scb$, then solving for $c$ yields that
\begin{equation}
 c = \sum_{i=0}^n(b^i-a_i)a_i. \label{eqn:cfixedpoint} 
\end{equation}
Thus, for a given base $b$ and an arbitrary positive integer $a$, there is at most one augmenting constant, $c$, such that $a$ is a fixed point of $\Scb$. 




\section{Fixed point oases}\label{S:OasisB}



We begin by defining the key concept in this paper, a $k$-oasis base $b$, which is analogous to the concept of a $k$-desert base $b$, defined in Part I of this paper~\cite{desert}.

\begin{definition} For $b\geq 2$ and $k\in \ZZ^+$, a {\em $k$-oasis base $b$} is a set of $k$ consecutive non-negative integers $c$ for each of which $\Scb$ has at least one fixed point.  An {\em oasis base $b$} is a $k$-oasis base $b$, for some $k \geq 1$.  The \emph{length} of a $k$-oasis base $b$ is $k$. \end{definition}


Theorem~\ref{oases} provides some basic facts about the existence and lengths of oases base $b$ for different values of $b \geq 2$. 


\begin{theorem}\label{oases}  
Let $b\geq 2$.
\begin{enumerate}
\item
There exists an oasis base $b$.
\item
If $b\geq 2$ is odd, then every $k$-oasis base $b$ has $k = 1$.
\item
If $b \geq 6$ is even, then there exists a $5$-oasis base $b$.
\end{enumerate}
\end{theorem}

\begin{proof}
First, for any base $b\geq 2$, since $S_{[0,b]}(1) = 1$, $\{0\}$ is a 1-oasis base $b$.

Next, let $b\geq 2$ be odd.
As shown by Baker Swart et al.~\cite[Lemma 2.3]{desert}, if $\Scb$ has a fixed point, then $c$ is even.  Part 2 of the theorem follows immediately.

Finally, let $b \geq 6$ be even and let $B = b/2$.  Set
\begin{align*}
 a_1&amp;={\left(B-2\right)b+1},  &amp;
c_1&amp;=B^2-4,\\
a_2&amp;={\left(B-1\right)b+2}, &amp; c_2&amp;=B^2-3, \\ 
a_3&amp;={Bb+2},   &amp; c_3&amp;=B^2-2, \\
a_4&amp;={\left(B-1\right)b+1},  &amp; c_4&amp;=B^2-1,\\
a_5&amp;={Bb+1},   &amp; c_5&amp;=B^2. 
\end{align*}
A direct calculation shows that for each $1 \leq i\leq 5$, 
$S_{[c_i,b]}(a_i) = a_i$.  Hence $\{c_1,c_2,c_3,c_4,c_5\}$ is a $5$-oasis base $b$.
\end{proof}

Given a $k$-oasis base $b$, it is easy to produce additional $k$-oases base $b$.

\begin{theorem}\label{t:infinite} Let $b \geq 2$ and $k \geq 1$.  If there exists a $k$-oasis base $b$, there exist infinitely many $k$-oases base $b$.
\end{theorem}

\begin{proof}
Fix $b\geq 2$ and let $\{c + j|1\leq j\leq k\}$ be a $k$-oasis base $b$.
For each $j$, $1\leq j \leq k$, let $a(j)$ denote a fixed point of $S_{[c+j,b]}$.
Fix $n \in \ZZ^+$ such that for each $j$, $a(j) &lt; b^n$.  Then for each positive integer $t$ and for each $j$, $t b^n + a(j)$ is a fixed point of 
\[S_{[c + j + t b^n - S_{[0,b]}(t), b]}.\] 
Hence for each $t\in \ZZ^+$, 
$\{c + j + t b^n - S_{[0,b]}(t)|1\leq j\leq k\}$ is a $k$-oasis base $b$.
\end{proof}

The following theorems provide properties of fixed points associated with values of $c$ that are in the same oasis base $b$.  
 Theorem~\ref{t:samedigit} provides that unless two such fixed points are quite small, they must have the same number of digits, and Theorem~\ref{T:matchingdigits} says that few of the digits of the fixed points may differ.
These theorems are used in proving Theorem~\ref{t:maxoasis}, which gives a general upper bound for the length of an oasis base $b$, and again in Section~\ref{S:MirageB}, in proving the correctness of a method we provide for improving this bound.


\begin{theorem}\label{t:samedigit}
Let $c \geq 0$ and $b \geq 2$.  Let $a \in \ZZ^+$ have $n + 1 &gt; 3$ digits and satisfy $\Scb({a}) = {a}$.  Then every fixed point with an augmenting constant in the same oasis base $b$ as $c$ has exactly $n+1$ digits.
\end{theorem}
 
We note that Theorem~\ref{t:samedigit} is optimal in that fixed points of two and three digits, respectively, can have augmenting constants in the same oasis.  For example, in base 16, the two-digit number $a = 85_{(16)}$ has augmenting constant $c = 44$ and the three-digit number $\hat{a} = 10(15)_{(16)} = 
1\cdot 16^2 + 15$ has augmenting constant $\hat{c} = 45$. Clearly $c$ and $\hat{c}$ are in the same oasis base 16.

\begin{proof}[Proof of Theorem~\ref{t:samedigit}]
Consider the collection of all values of $c$ in a fixed oasis base $b$, and the set of all fixed points of the happy functions base $b$ with those $c$s as augmenting constants.  Assume that two of these fixed points have different numbers of digits and at least one of them has more than 3 digits.
Then there must exist augmenting constants $\bar c$ and $\hat c$ in the oasis with $|\bar c - \hat c| \leq 1$, and fixed points $\bar a$ of $S_{[\bar c,b]}$ with $\bar n + 1 &gt; 3$ digits and $\hat a$ of $S_{[\hat c,b]}$ with $\hat n + 1 \neq \bar n + 1$ digits.  We may assume without loss of generality that $\hat n &lt; \bar n$.

In Part I of this paper~\cite[Theorem 4.2]{desert}, Baker Swart et al.\ showed that for $n \geq 2$, if $\Scb$ has a fixed point of $n + 1$ digits, then $m_{b,n} \leq c \leq M_{b,n}$, where the bounds, which are given explicitly in terms of their parameters, are sharp. They also showed~\cite[Lemma 4.3]{desert} that, for $n \geq 2$, $M_{b,n} + 1 &lt; m_{b,n+1}$.

In the same work~\cite[Theorem 4.2]{desert}, the authors prove that, for $n \geq 2$, if $\Scb$ has a fixed point of $n + 1$ digits, then $m_{b,n} \leq c \leq M_{b,n}$, where the bounds, which are given explicitly in terms of their parameters, are sharp.  Additionally, they show~\cite[Lemma 4.3]{desert} that, for $n \geq 2$, $M_{b,n} + 1 &lt; m_{b,n+1}$. It follows (using a simple induction argument) that if $\hat n \geq 2$, then $\hat c + 1\leq M_{b,\hat n} + 1 &lt; m_{b,\bar n} \leq \bar c$. But this implies that $1 &lt; \bar c - \hat c = |\bar c - \hat c|$, a contradiction.  

Thus, we have that $\hat n &lt; 2$. Letting 
$\hat a = \sum_{i=0}^{\hat n}a_ib^i$, Equation~(\ref{eqn:cfixedpoint}) yields
\[\hat c = (b - a_1)a_1 + (1-a_0)a_0.\]
The largest possible value of $(b-a_1)a_1$ occurs when 
$a_1 = \lfloor b/2\rfloor$ and the  largest possible value of $(1-a_0)a_0$ is 0.  Thus,
\[\hat c = (b - a_1)a_1 + (1-a_0)a_0 \leq (b - \lfloor b/2\rfloor) \lfloor b/2\rfloor + 0 \leq b^2/4.\] 
Since $\bar a$ has more than three digits, $\bar n \geq 3$, and so~\cite[Theorem 4.2]{desert} implies that
\[\bar{c} \geq m_{b,\hat{n}} = b^{\hat{n}} - b^2 + 3b  - 3 \geq b^3 - b^2 + 3b - 3.\]
Combining these and the fact that $b \geq 2$ yields
\[1 \geq |\bar c - \hat c| \geq (b^3 - b^2 + 3b - 3) - b^2/4 &gt; 1,\] a contradiction, completing the proof.
\end{proof} 


\begin{theorem}\label{T:matchingdigits} Fix $b \geq 2$ and let $c$ and $\hat c$ be in the same oasis base $b$.  Let $\Scb(a) = a$ and $S_{[\hat{c},b]}(\hat{a})=\hat{a}$.  Then, letting $a_i$ and $\hat{a}_i$ denote the coefficients of $b^i$ in the base $b$ expansions of $a$ and $\hat{a}$, respectively, for each $i \geq 3$, $a_i = \hat{a}_i$. \end{theorem}


\begin{proof}
Suppose for a contradiction that there exists an $i\geq 3$ such that $a_i \neq \hat{a}_i$. 
Then, at least one of $a$ and $\hat{a}$ has more than three digits, and so, by Theorem~\ref{t:samedigit}, $a$ and $\hat{a}$ have the same number of digits, say $n + 1 &gt; 3$.  We may assume, without loss of generality, that $|\hat{c} - c| \leq 1$.

Fix $j\geq 3$ maximal such that $a_j \neq \hat{a}_j$.
 Then using Equation~(\ref{eqn:cfixedpoint}), we have
\begin{align*}
|\hat{c} - c| 
&amp; = \left|\left(\sum_{i=0}^n(b^i-\hat{a}_i)\hat{a}_i\right) - \left(\sum_{i=0}^n(b^i-a_i)a_i\right)\right|\\
&amp; = \left| \sum_{i=0}^j\left((b^i-\hat{a}_i)\hat{a}_i - (b^i-a_i)a_i\right)\right|\\
&amp; \geq \left| (b^j-\hat{a}_j)\hat{a}_j - (b^j - a_j)a_j \right| -
\sum_{i=0}^{j-1}\left|(b^i - \hat{a}_i)\hat{a}_i - (b^i-a_i)a_i\right|.
\end{align*}

For $i\geq 2$ and $0 \leq x \leq b-1$, the first derivative of the function $f(x) = (b^i - x)x$ is positive and decreasing.  Thus, the function is increasing at a decreasing rate over the domain.  Therefore, the smallest difference between the function values for two integer values of $x$ occurs when $x = b - 1$ and $x = b - 2$.  Similarly, the largest difference occurs when $x = b - 1$ and $x = 0$.  It follows that
\begin{align*}
|\hat{c} - c| 
\geq{}&amp;\left((b^j - (b - 1))(b - 1) - (b^j - (b - 2))(b - 2)\right) \\
&amp; - \sum_{i=2}^{j-1}\left((b^i - (b - 1))(b - 1) - (b^i - 0)(0)\right) \\
&amp; - \sum_{i=0}^{1}\left|(b^i - \hat{a}_i)\hat{a}_i - (b^i-a_i)a_i\right|\\
\geq{}&amp; b^j - (b-1)^2 + (b-2)^2 - \sum^{j-1}_{i=2} b^i(b-1) + (j - 2)(b - 1)^2 \\
&amp; - \left|\left(b-\frac{b}{2}\right)\left(\frac{b}{2}\right) - (b - 0)(0)\right| \\
&amp; -|(1 - (b - 1))(b - 1) - (1 - 0)(0)|\\
={}&amp; b^2 + (b - 2)^2 + (j - 3)(b - 1)^2 - \frac{b^2}{4} - (b - 1)(b - 2).
\end{align*}
Since $j\geq 3$ and $b \geq 2$, this implies that
\[1 \geq |\hat{c} - c| \geq \frac{3}{4}b^2 - b + 2 &gt; 2,\] a contradiction.
Thus no such $i \geq 3$ exists, as desired.
\end{proof}

\begin{theorem}\label{t:maxoasis}
Let $b \geq 2$.  If there exists a $k$-oasis base $b$, then 
\begin{equation*}
k \leq \frac{b^3}{2} +\frac{b^2}{2} - b.
\end{equation*}
\end{theorem}

\begin{proof}
If $b$ is odd, then, by Theorem~\ref{oases}, every oasis base $b$ has length 1, which is less than $b^3/2 + b^2/2 - b$.  So we assume that $b$ is even.

Let $\{c + j | 1 \leq j \leq k\}$ be a $k$-oasis base $b$.  For each $j$, $1 \leq j \leq k$, let $a(j)$ be a fixed point of $S_{[c+j,b]}$. By Theorem~\ref{T:matchingdigits}, the fixed points, $a(j)$, differ in, at most, the rightmost three digits.  Since each fixed point corresponds to exactly one augmenting constant, this implies an initial bound: $k \leq b^3$.  But we can improve on this.


Baker Swart et al.~\cite[Theorem 2.1]{desert} prove that if $a(j)$ is a multiple of $b$, then $a(j) + 1$ is also a fixed point of $S_{[c+j,b]}$.  Thus, substituting $a(j) + 1$ for $a(j)$, if necessary, we may assume that none of the $a(j)$ is a multiple of $b$.  This leaves us with $b-1$ possible rightmost digits.

Similarly, if $a(j)$ has second rightmost digit equal to $d \neq 0$, then the number obtained by replacing that digit with the digit $b-d$ is another fixed point of $S_{[c+j,b]}$~\cite[Lemma 2.2]{desert}.  Thus we may assume that none of the $a_i$ have second rightmost digit greater than $b/2$.  This leaves us with $(b+2)/2$ possible second rightmost digits.  

So, the number of possible values of the rightmost three digits of the $a(j)$s is 
\[(b)\left(\frac{b+2}{2}\right)(b - 1).\]
Since, for each of the $k$ augmenting constants, $c + j$, there is a distinct fixed point, $a(j)$, we conclude that the number of augmenting constants in the oasis is
\begin{equation*}
k \leq (b)\left(\frac{b+2}{2}\right)(b - 1) = \frac{b^3}{2}+\frac{b^2}{2}-b,
\end{equation*}
as desired.
\end{proof}

\section{Maximal lengths of oases base \texorpdfstring{$b$}{b}}
\label{S:MirageB}


In the previous section, we determined a general formula for an upper bound for the length of an oasis base $b$, for $b\geq 2$.  In this section, we present an algorithm for determining a new bound on this length, which, in many cases, can be shown to provide the precise maximal length of oases base $b$. 


 From Theorem~\ref{oases}, we know that if $b$ is odd, every oasis base $b$ has length 1. For bounding the oasis lengths for even bases, we introduce the concept of a mirage base $b$.  
For convenience, we extend the domain of the function $S_{[0,b]}$ for $b \geq 2$ to include 0 by defining $S_{[0,b]}(0) = 0$.  

\begin{definition} For $b\geq 2$ and $k \in \ZZ^+$, a {\em $k$-mirage base $b$} is a set of $k$ consecutive integers $\{d_1,..., d_k\}$, such that for each $1\leq i \leq k$, $d_i = r_i - S_{[0,b]}(r_i)$ with $r_i$ a non-negative integer of at most three digits.  A {\em mirage base $b$} is a $k$-mirage base $b$ for some $k \geq 1$. \end{definition}  

A mirage base $b$ may or may not be an oasis base $b$.  
We first provide a large class of mirages that actually are oases.

\begin{lemma}\label{l:mirageisoasis}
If a $k$-mirage base $b$ contains only positive integers, then it is a $k$-oasis base $b$. 
\end{lemma}

\begin{proof}
Given a $k$-mirage containing only positive integers, using the notation in the definition of $k$-mirage base $b$, for each $1\leq i \leq k$, we have $S_{[d_i,b]}(r_i) = d_i + S_{[0,b]}(r_i) = r_i$.  Thus $r_i$ is a fixed point of $S_{[d_i,b]}$, and so $\{d_1,..., d_k\}$ is a $k$-oasis base $b$.
\end{proof}

Of course, not all mirages base $b$ are oases base $b$.  For example, 
$-4 = 16 - S_{[0,6]}(16)$, $-3 = 22 - S_{[0,6]}(22)$, $-2 = 2 - S_{[0,6]}(2)$, $-1 = 9 - S_{[0,6]}(9)$, and $0 = 1 - S_{[0,6]}(1)$, implying that $\{-4,-3,-2,-1,0\}$ is a 5-mirage base 6, though not an oasis base 6.

We next show that given a $k$-oasis base $b$, there must exist a $k$-mirage base $b$.

\begin{theorem}\label{t:mirage}
Given $b\geq 2$ and $k\in \ZZ^+$, if there exists a $k$-oasis base $b$, then there exists a $k$-mirage base $b$.
\end{theorem}
 
\begin{proof}
Let $\mathcal O = \{c + j| 1\leq j\leq k\}$ be a $k$-oasis base $b$, and for each $1\leq j \leq k$, let $a(j) \in \ZZ^+$ be a fixed point of $S_{[c +j,b]}$.  

First, consider the case in which each $a(j)$ has 3 or fewer digits.  Then for $1\leq j \leq k$, $a(j) = S_{[c+j,b]}(a(j)) = (c + j) + S_{[0,b]}(a(j))$, and so $c + j = a(j) - S_{[0,b]}(a(j))$.  Thus $\mathcal O$ is a $k$-mirage base $b$, and we are done.

Next, consider the case in which, for at least one value of $j$, $a(j)$ has more than 3 digits. Then, by Theorem~\ref{t:samedigit}, all of the $a(j)$ have the same number of digits, say $n + 1 &gt; 3$.  For each $1\leq j\leq k$, let $0\leq a(j)_i \leq b-1$, such that 
\[a(j) = \sum_{i=0}^n a(j)_ib^i,\]  
and define \[r_j = \sum_{i=0}^2 a(j)_ib^i.\] 
By Theorem~\ref{T:matchingdigits}, for each $1\leq j\leq k$ and $i \geq 3$, $a(j)_i = a(1)_i$.  Thus, for each $1\leq j\leq k$, 
\begin{equation} \label{eq:C}
a(j) = r_j + \sum_{i=3}^n a(j)_ib^i = r_j + \sum_{j=3}^n a(1)_ib^i = r_j + a(1) - r_1.
\end{equation}
Further, since $a(j)$ is a fixed point of $S_{[c+j,b]}$, we have that
\begin{equation}\label{eq:C-hat}
\begin{split}
a(j) &amp;= S_{[c+j,b]}(a(j)) = (c + j) + S_{[0,b]}(r_j) + \sum_{i=3}^n a(j)_i^2   \\
&amp;= (c + j) + S_{[0,b]}(r_j) + \sum_{i=3}^n a(1)_i^2. 
\end{split}
\end{equation} 
Thus, using equations~(\ref{eq:C}) and~(\ref{eq:C-hat}), for each $1\leq j \leq k$,
\begin{equation} \label{eq:consec}
r_j - S_{[0,b]}(r_j) = \left(r_1 - a(1) + \sum_{i=3}^n a(1)_i^2 + c\right) + j.
\end{equation} 
Since the only value on the right-hand-side of Equation~(\ref{eq:consec}) dependent on $j$ is $j$ itself,  
\[\left\{r_j - S_{[0,b]}(r_j) | 1\leq j \leq k\right\}\]
is a set of consecutive integers and thus is a $k$-mirage base $b$.
\end{proof}


It follows from Theorem~\ref{t:mirage} that the length of the longest mirage base $b$ bounds the length of the longest oasis base $b$. Since each element in a mirage is generated by a fixed point between $0$ and $b^3$ exclusive, the maximum length of a mirage base $b$ can be determined by a direct computer search.  Formalizing this algorithm: in order to determine, for some fixed $b \geq 2$, the maximal length of a mirage base $b$, and to check whether this is necessarily equal to the maximal length of an oasis base $b$, the following steps suffice.
\begin{enumerate}
\item
For each $0 &lt; r &lt; b^3$, compute $d = r - S_{[0,b]}(r)\in \ZZ$.
\item
Sort the values of $d$.
\item
Determine the length of the longest string of consecutive values of $d$.
\item
Check whether there is a longest string in which all of the values of $d$ are positive.
\end{enumerate}
The result of step 3 is the maximal length of a
mirage base $b$ and, therefore, a bound on the length of the maximal length oasis base $b$.  Each longest string found in step 3 is an example of a maximal length mirage base $b$.  If step 4 is answered in the affirmative, then this string of positive values of $d$ is also an example of a maximal length oasis base $b$.

We carry out this algorithm for all even bases $2 \leq b \leq 20$, in each case finding a maximal oasis base $b$.  We summarize the results in the following theorem.

\begin{theorem}
The maximal lengths of oases base $b$ for bases 2, 4, and 6, are 2, 6, and 5, respectively; the maximal length of oases base $b$ for bases 8, 10, 12, 14, 16, and 18 is 8; and the maximal length of oases base 20 is 9.
\end{theorem}

In Table~\ref{max} we provide, for each even base, $2\leq b \leq 20$,
the minimal example of an oasis of maximal length, along with the smallest
fixed point of the augmented happy function determined by each augmenting
constant in the oasis.
\begin{table}[H]
\begin{center}
{\small
\begin{tabular}{|c|c|c|}\hline 
Base&amp;Length&amp;Minimal maximal length oasis  \\
\hline
&amp;&amp;Smallest fixed points\\
\hline\hline
2&amp; 2 &amp; $\{3,4\}$\\
\hline
&amp; &amp;  ${4,6}$\\
\hline\hline
4&amp; 6 
&amp; $\{28,29,30,31,32,33\}$ \\
\hline
&amp;
&amp; ${32, 38, 42, 36, 40, 51}$\\
\hline\hline
6&amp; 5  
 &amp; $\{5,6,7,8,9 \}$ \\
\hline
&amp;
 &amp;  ${6, 14, 20, 12, 18}$\\
\hline\hline
8 &amp; 8 
 &amp; $\{304,305,306,307,308,309,310,311\}$  \\
\hline
&amp;
 &amp;  ${347, 338, 391, 336, 346, 354, 344, 352}$\\
\hline\hline
10 &amp; 8 
 &amp; $\{487, 488, 489, 490, 491, 492, 493, 494\}$ \\
\hline
&amp;
 &amp;  ${544, 554, 522, 533, 520, 609, 543, 532 }$\\
\hline\hline
12 &amp; 8 &amp; \{172, 173, 174, 175, 176, 177, 178, 179\}  \\
\hline
&amp; &amp; {207, 194, 299, 192, 206, 218, 204, 216}  \\
\hline\hline
14 &amp; 8
 &amp; $\{421, 422, 423, 424, 425, 426, 427, 428\}$ \\
\hline
&amp;
 &amp;  ${434, 451, 601, 480, 494, 450, 465, 448}$\\
\hline\hline
16 &amp; 8
 &amp; $\{559, 560, 561, 562, 563, 564, 565, 566\}$ \\
\hline
&amp;
 &amp;  ${628, 644, 594, 611, 592, 799, 627, 610}$\\
\hline\hline
18 &amp; 8
 &amp; $\{1663, 1664, 1665, 1666, 1667, 1668, 1669, 1670\}$ \\
\hline
&amp;
 &amp;  ${1768, 1786, 1730, 1749, 1728, 1960, 1767, 1748}$\\
\hline\hline
20 &amp; 9
 &amp; $\{5124, 5125, 5126, 5127, 5128, 5129, 5130, 5131, 5132\}$ \\
\hline
&amp;
 &amp;  ${5383, 5362, 5699, 5360, 5382, 5402, 5380, 5400, 5617}$\\
 \hline
\end{tabular}
\caption{Minimal valued maximal length oases and smallest fixed points for small even bases. (Results are 
all given in base 10.) \label{max}}
}
\end{center}
\end{table}



\begin{thebibliography}{9}

\bibitem{augment}
B.~Baker~Swart, K.~Beck, S.~Crook, C.~Eubanks-Turner, H.~Grundman, M.~Mei,  and 
  L.~Zack, Augmented generalized happy functions, {\it Rocky Mountain J. Math.}, \textbf{47} (2017), 403--417.

\bibitem{desert}
B.~Baker~Swart, K.~Beck, S.~Crook, C.~Eubanks-Turner, H.~Grundman, M.~Mei,  and
  L.~Zack, Fixed points of augmented generalized happy functions, {\it Rocky Mountain J. Math.}, \textbf{48} (2018), 47--58.

\bibitem{genhappy}
H.~G. Grundman and E.~A. Teeple, Generalized happy numbers, {\it Fibonacci
  Quart.}, \textbf{39} (2001), 462--466.

\bibitem{guy}
Richard~K. Guy, \emph{Unsolved Problems in Number Theory}, 3rd ed.,
Springer-Verlag, 2004.

\bibitem{honsberger}
Ross Honsberger, \emph{Ingenuity in Mathematics},
New Mathematical Library, Vol.~23,
 Random House, Inc., 1970.

\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2010 \textit{Mathematics Subject Classification:} Primary 11A63.

\noindent \emph{Keywords:} happy number, fixed point, iteration.


\bigskip
\hrule
\bigskip

\noindent (Concerned with sequence
\seqnum{A007770}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received February 8 2019; 
revised version received July 31 2019. 
Published in {\it Journal of Integer Sequences}, August 23 2019.

\bigskip
\hrule
\bigskip

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


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