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\begin{center}
\vskip 1cm{\LARGE\bf 
Generalized Anti-Waring Numbers
}
\vskip 1cm
\large
Chris Fuller and Robert H. Nichols, Jr.\\ 
Labry School of Science, Technology, and Business\\ 
Cumberland University\\ 
1 Cumberland Square\\ 
Lebanon, TN 37087\\ 
USA \\ 
\href{mailto:cfuller@cumberland.edu}{\tt cfuller@cumberland.edu}\\ 
\href{mailto:rnichols@cumberland.edu}{\tt rnichols@cumberland.edu}
\end{center}

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\begin{abstract}
The anti-Waring problem considers the smallest positive integer such
that it and every subsequent integer can be expressed as the sum of the
$k^{\rm th}$ powers of $r$ or more distinct natural numbers.  We give a
generalization that allows elements from any nondecreasing sequence,
rather than only the natural numbers. This generalization is an
extension of the anti-Waring problem, as well as the idea of complete
sequences.   We present new anti-Waring and generalized anti-Waring
numbers, as well as a result to verify computationally when a
generalized anti-Waring number has been found.
\end{abstract}

\section{Introduction}

For positive integers $k$ and $r$, the anti-Waring number $N(k,r)$ is
defined to be the smallest positive integer such that $N(k,r)$ and
every subsequent positive integer can be expressed as the sum of the
$k^{\rm th}$ powers of $r$ or more distinct positive integers.  Several
authors \cite{Deering,FPV,Johnson,Looper} recently reported results on
anti-Waring numbers.

Early results considered only $r=1$.  As early as 1948, Sprague found that $N(2,1)=129$ \cite{Sprague1} and proved that $N(k,1)$ exists for  all $k\geq 2$ \cite{Sprague2}.  In 1964, Graham \cite{Graham31} reported that $N(3,1)=12759$ (Graham \cite{Graham31} references another Graham paper ``On the Threshold of completeness for certain sequences of polynomial values'' said to appear circa 1964).  Dressler and Parker \cite{Dressler} also computed $N(3,1)$ in 1974.  Lin \cite{Lin} used Graham's method to find that $N(4,1) = 5134241$ with a computer in 1970.  In 1992, Patterson \cite[pp.\ 18--23]{Patterson} found that $N(5,1)=67898772$.  In this paper, we independently verify each of these numbers and show that $N(6,1)=11146309948$.

More recently, Looper and Saritzky \cite{Looper} proved that $N(k,r)$ exists for all positive integers $k$ and $r$.  Deering and Jamieson \cite{Deering} found specific values of $N(2,r)$ for $1\leq r\leq 10$ and $N(3,r)$ for $1\leq r\leq 5$.  Shortly afterwards, Fuller et al.\ \cite{FPV}  computed values of $N(2,r)$ for $1\leq r\leq 50$ and $N(3,r)$ for $1\leq r\leq 30$.  We also verify these numbers and present $N(k,r)$ for more values of $k$ and $r$.  One can verify a suspected value of $N(k,r)$ using different sets of conditions \cite{Deering,FPV}.

In an effort to generalize the anti-Waring results we consider a nondecreasing sequence of positive integers $A=(a_i)_{i\in\mathbb{N}}$.  Here and throughout we use $\mathbb{N}=\left\{1,2,3,\dots \right\}$.  For positive integers $k$, $n$, and $r$ we define the \emph{generalized anti-Waring number} $N(k,n,r,A)$ to be the smallest positive integer, if it exists, such that it and every subsequent positive integer can be expressed as the sum of the $k^{\rm th}$ powers of the $a_i$ with $i\geq n$ ranging over $r$ or more distinct values.  If the sequence $A$ has all distinct elements, we may use set notation for the last argument of the generalized anti-Waring number.  The generalized anti-Waring number $N(k,n,r,A)$ does not exist for all sequences $A$ (see Theorems~\ref{dne_gcd} and \ref{dne_factorial} in Section~\ref{section:NknrA}).  Looper and Saritzky \cite{Looper} proved that both the anti-Waring number $N(k,r)$ and the generalized anti-Waring number $N(k,n,r,\mathbb{N})$ exist for all positive integers $k$, $n$, and $r$.

Early results of these generalized anti-Waring numbers when restricting $r$ to 1 used different terminology.  A nondecreasing sequence $S$ of positive integers is \textit{complete} if all sufficiently large positive integers can be written as a sum of distinct elements of $S$. If $S$ is a complete sequence, the \textit{threshold of completeness}, $\theta (S)$, is the largest positive integer that is not expressible as a sum of distinct elements of $S$.  Therefore, the threshold of completeness, $\theta (S)$, is one less than the generalized anti-Waring number $N(1,1,1,S)$.  Also, if $S=(s_i)_{i\in\mathbb{N}}$ is a nondecreasing sequence of positive integers such that the sequence $(s_i^k)_{i\geq n}$ is complete, then the generalized anti-Waring number $N(k,n,1,S)$ exists and $N(k,n,1,S) -1 = \theta\left( (s_i^k)_{i\geq n}\right)$.  Brown \cite{Brown} defined a sequence to be complete only when the threshold of completeness is zero; we use the more general definition.

In the literature on complete sequences, some authors only report that a sequence is complete and hence the generalized anti-Waring number exists; some authors actually find the threshold of completeness.  In 1952, Lekkerkerker \cite{Lekkerkerker} reported an account of the Zeckendorf representation (circa 1939 \cite{Zeckendorf}), i.e.,  that every natural number is either a Fibonacci number or can be expressed as the sum of nonconsecutive Fibonacci numbers.  Hence the generalized anti-Waring number for the Fibonacci sequence $F$ is $N(1,1,1,F)=1$.  In 1975, Kl{\o}ve \cite{Klove} found thresholds of completeness for sequences of the form $\left( \lfloor i^{\alpha} \rfloor\right)_{i\in\mathbb{N}}$, where $\lfloor x\rfloor$ is the floor function, for $1\leq\alpha\leq 4.18$ in increments of 0.02.  In 1978, Porubsk\'{y} \cite{Porubsky} proved that $N(k,1,1,\mathbb{P})$ exists for all positive integers $k$ and the sequence of primes $\mathbb{P}$.  Burr and Erd\H{o}s \cite{BurrEr} considered perturbations of complete sequences that resulted in noncomplete sequences and vice versa.

Generalized anti-Waring numbers extend the concept of anti-Waring numbers to sequences other than $\mathbb{N}$.  The generalization also extends the concept of complete sequences to consider sums of $r$ or more terms.  We will present conditions needed to verify values of $N(k,n,r,A)$ computationally, sequences for which no $N(k,n,r,A)$ exists, and new values of $N(k,n,r,A)$ for various sequences.

\section{Verifying $N(k,n,r,A)$, when it exists} \label{section:NknrA}

For given positive integers $k$, $n$, $r$, and any nondecreasing sequence of positive integers $A=(a_i)_{i\in\mathbb{N}}$, we define a positive integer to be \textit{$(k,n,r,A)$-good} if it can be written as a sum of the $k^{\rm th}$ powers of $r$ or more distinct elements of the sequence $(a_i)_{i\geq n}$.  We define a positive integer that is not $(k,n,r,A)$-good to be \textit{$(k,n,r,A)$-bad}.  Hence the generalized anti-Waring number $N(k,n,r,A)$ is the smallest positive integer such that it and every subsequent integer is $(k,n,r,A)$-good.  Equivalently the threshold of completeness $N(k,n,r,A)-1$ is the largest integer that is $(k,n,r,A)$-bad.

The generalized anti-Waring number $N(k,n,r,A)$ does not exist for all sequences $A$.  For example, the sum of any elements of the sequence $(2,4,6,8, \ldots )$ of positive even integers will never be odd.  This is an instance of a more general phenomenon.

\begin{theorem} \label{dne_gcd}
Let $A=(a_i)_{i\in\mathbb{N}}$ be a nondecreasing sequence of positive integers.  If all $a_i$ for $i\geq n$ have a common divisor $d&gt;1$, then for any positive integers $k$ and $r$, the generalized anti-Waring number $N(k,n,r,A)$ does not exist.
\end{theorem}
\begin{proof}
Every sum of positive powers of the $a_i$, $i\geq n$, is divisible by $d$.  Since $d&gt;1$, arbitrarily large integers not divisible by $d$ exist.  Thus, arbitrarily large integers not representable in any way as a sum of powers of some of the $a_n, a_{n+1}, \ldots$ also exist.
\end{proof}

If instead the greatest common divisor is one, then the generalized anti-Waring number may or may not exist.  We will consider examples of both cases.

As an additional example, the sequence of factorials has no generalized anti-Waring number.

\begin{theorem} \label{dne_factorial}
Let $A=(i!)_{i\in\mathbb{N}}$, and let $k$, $n$, and $r$ are any positive integers.  Then the generalized anti-Waring number $N(k,n,r,A)$ does not exist.
\end{theorem}
\begin{proof}
First notice that for each $a_i\in A$,
$$a_i^k \bmod 6\equiv \begin{cases} 1,&amp;\text{if $i=1$;} \\ 2^k \bmod 6,&amp;\text{if $i=2$;} \\0, &amp;\text{if $i&gt;2$.}\end{cases}$$
Consider any $(k,n,r,A)$-good number $m$.  Distinct integers $i_1, i_2, \ldots ,i_t$ exist such that 
$$m = a_{i_1}^k + a_{i_2}^k + \cdots + a_{i_t}^k$$
where $t\geq r$ and $i_\alpha \geq n$ for each $\alpha\in\{1,2,\ldots ,t\}$.  Thus the sum $m$ must be $0$, $1$, $2^k$, or $1+2^k$ modulo 6.  Since we can have at most four consecutive $(k,n,r,A)$-good integers, no largest $(k,n,r,A)$-bad integer exists. 
\end{proof}

On the other hand, in some cases the generalized anti-Waring number $N(k,n,r,A)$ is known to exist, but its value has not been found.  As mentioned above, both the anti-Waring number $N(k,r)$ and the generalized anti-Waring number $N(k,n,r,\mathbb{N})$ exist for all $k$, $n$, and $r$ \cite{Looper}.  A general formula for either of these is not known, but we present several values in the next section.  We rewrite the following result related to complete sequences by Brown \cite[Theorem 1]{Brown} in terms of generalized anti-Waring numbers.

\begin{theorem} \label{brownthm}
Let $k$ and $n$ be positive integers, and let $A=(a_i)_{i\in\mathbb{N}}$ be a nondecreasing sequence of positive integers.  The generalized anti-Waring number $N(k,n,1,A)$ both exists and equals one if and only if (i) $a_n = 1$ and (ii) for all integers $p \geq n$, $a_{p+1}^k\leq 1 + \sum_{i=n}^p a_{i}^k$.
\end{theorem}

This result only considers $r=1$.  Also since Brown \cite{Brown} defined complete sequences requiring the threshold of completeness to be zero, he requires $a_n=1$.  Theorem~\ref{brownthm} proves that all positive integers are representable as a sum of different elements of sequences such as the natural numbers, the Fibonacci numbers, and the powers of two (including $2^0$).  We must consider different conditions for the more general definition of complete sequences with any threshold of completeness.

The next result from Graham \cite[Theorem 4]{Graham31} establishes completeness conditions for sequences generated by polynomials.

\begin{theorem}\label{grahamthm}
Let $f(x)$ be a polynomial with real coefficients expressed in the form 
$$f(x)=\alpha_0+\alpha_1\binom{x}{1}+\cdots + \alpha_n\binom{x}{n},\quad \alpha_n\neq 0.$$
The sequence $S(f)=(f(1),f(2),\cdots )$ is complete if and only if 
	\begin{enumerate}
		\item $\alpha_k=p_k/q_k$ for some integers $p_k$ and $q_k$ with $\gcd(p_k,q_k) =1$ and $q_k\neq 0$ for $0\leq k\leq n$,
		\item $\alpha_n&gt;0$, and
		\item $\gcd (p_0,p_1,\ldots ,p_n)=1$.
	\end{enumerate}
\end{theorem}

Again, in terms of generalized anti-Waring numbers Theorem~\ref{grahamthm} only considers the case of $r=1$ and can only be used to establish that a given generalized anti-Waring number exists.  As a remark to this theorem, Graham notes that a sequence $(f(1),f(2),f(3),\ldots )$ is complete if and only if $(f(n),f(n+1),f(n+2),\ldots )$ is complete for any $n$.  The next theorem shows that nothing like this can be expected in general.

\begin{theorem}
Let $k$, $n$, and $r$ be positive integers, and let $A$ be a sequence of nondecreasing positive integers.  If the generalized anti-Waring number $N(k,n,r,A)$ exists, then so does $N(k,j,r,A)$ for $j\in\{1,2,\ldots , n-1\}$ and $N(k,j,r,A)\leq N(k,n,r,A)$.  Furthermore, the converse is false.
\end{theorem}
\begin{proof}
The implication is clear.  If all positive integers greater than or equal to $N(k,n,r,A)$ can be written as a sum $k^{\rm th}$ powers of $r$ or more distinct elements of $(a_i)_{i\geq n}$, then, with the same elements, each positive integer can be written as a sum $k^{\rm th}$ powers of $r$ or more distinct elements of $(a_i)_{i\geq j}$ for $j\in\{1,2,\ldots , n-1\}$.  Therefore, we have $N(k,j,r,A)\leq N(k,n,r,A)$ for $j\in\{1,2,\ldots , n-1\}$.

To see that the converse is false, consider the sequence $A=\left(2^{i-1}\right)_{i\in\mathbb{N}}$.  From the binary representation of the positive integers, the generalized anti-Waring number $N(1,1,1,A)$ clearly exists and equals one.  However, the generalized anti-Waring number $N(1,2,1,A)$ does not exist because no odd integer can be expressed as a sum of elements from $\left(2^{i-1}\right)_{i\geq 2}$.
\end{proof}


In general, whether $N(k,n,r,A)$ exists or not cannot easily be determined.  However, we can validate a suspect value of $N(k,n,r,A)$ if enough consecutive integers are $(k,n,r,A)$-good and certain other conditions are met.  Theorem~\ref{verify_N} is a generalization of a recent result for anti-Waring numbers \cite[Theorem 2.2]{FPV}.


\begin{theorem} \label{verify_N}
Let $k$, $n$, $r$, $b$, and $\hat{N}$ be positive integers, and let $A=(a_i)_{i\in\mathbb{N}}$ with $0&lt;a_i\leq a_{i+1}$ and $a_i\in\mathbb{N}$ for all $i$.  If the consecutive integers $\{\hat{N},\ldots ,b^k\}$ are all $(k,n,r,A)$-good, the number $\hat{N}-1$ is $(k,n,r,A)$-bad, and there exists a positive integer $x$ such that the conditions
	\begin{enumerate}
	\item \label{condA} $\hat{N}\leq b^k+1-(b-x)^k $,
	\item \label{condB} $a_n\leq b-x$,
	\item \label{condD} $\displaystyle 0&lt;\left(\sum_{i=n}^{n+r-2}a_i^k\right)+2(m-x)^k-(m+1)^k$ for all $m\geq b$, and
	\item \label{condE} $(m+1)^k-(m-x)^k\leq m^k$ for all $m\geq b$
	\end{enumerate}
hold, then the generalized anti-Waring number $N(k,n,r,A)$ exists and equals $\hat{N}$. Note:  The sum in condition~\ref{condD} is zero if $r=1$.
\end{theorem}
\begin{proof}
We want to prove that if $\ell\leq m^k$ and $\ell$ is $(k,n,r,A)$-bad, then $\ell\leq\hat{N}-1$ by induction on $m$ with $m\geq b$.  

This is clearly true for $m=b$ as we know the consecutive integers $\{\hat{N},\ldots, b^k\}$ are all $(k,n,r,A)$-good.

Now suppose $\ell\leq(m+1)^k$ and $\ell$ is $(k,n,r,A)$-bad.  If $\ell\leq m^k$, then by induction $\ell\leq\hat{N}-1$.  Next, consider $\ell$ such that
\begin{equation}\label{ellBound}m^k+1\leq \ell\leq (m+1)^k.\end{equation}
Notice $b^k-(b-x)^k\leq m^k-(m-x)^k$ for $m\geq b$.  Using this along with \eqref{ellBound} and condition~\ref{condA}, we have
\begin{equation}\label{ellCap}\hat{N}\leq\ell-(m-x)^k.\end{equation}
To see that $\ell-(m-x)^k$ is $(k,n,r,A)$-bad, suppose it is $(k,n,r,A)$-good.  Then
$$\ell-(m-x)^k=a_{i_1}^k+a_{i_2}^k+a_{i_3}^k+\cdots+a_{i_t}^k$$
where $t\geq r$, $i_\alpha\neq i_\beta$ for all $\alpha\neq\beta$, and $i_\alpha\geq n$ for all $\alpha\in\{1,2,\ldots, t\}$.  Since $\ell$ is $(k,n,r,A)$-bad and
$$\ell=a_{i_1}^k+a_{i_2}^k+a_{i_3}^k+\cdots+a_{i_t}^k+(m-x)^k,$$
either $m-x&lt;a_n$, which contradicts condition~\ref{condB}, or $a_{i_\alpha}=m-x$ for some $\alpha\in\{1,2,\ldots ,t\}$.  Therefore,
$$\ell\geq a_{n}^k+a_{n+1}^k+a_{n+2}^k+\cdots+a_{n+r-2}^k+2(m-x)^k.$$
If $r=1$, this is just $\ell\geq 2(m-x)^k$.  Combining with \eqref{ellBound}, we get
$$\left(\sum_{i=n}^{n+r-2}a_i^k\right)+2(m-x)^k-(m+1)^k \leq 0.$$
This contradicts condition~\ref{condD} and means that $\ell-(m-x)^k$ must be $(k,n,r,A)$-bad.

Now from \eqref{ellBound} and condition~\ref{condE},
$$\ell-(m-x)^k\leq (m+1)^k-(m-x)^k\leq m^k.$$
By induction we then have $\ell-(m-x)^k\leq\hat{N}-1$.  This contradicts \eqref{ellCap}.  Hence there are no $\ell$ that are $(k,n,r,A)$-bad and satisfy \eqref{ellBound}.

\end{proof}

Most of the threshold of completeness results in the literature of complete sequences rely on work by Richert \cite{Richert}, where different sufficient conditions imply that a sequence is complete when restricting $r=1$.  Our algorithms for computing generalized anti-Waring numbers were designed to stop when $x$ and $b$ are found satisfying Theorem~\ref{verify_N}.
 
\section{Values of $N(k,n,r,A)$} \label{secValues}

As a result of Theorems~\ref{dne_gcd} and \ref{dne_factorial}, we know that $N(k,n,r,A)$ does not exist for all values of $k$, $n$, and $r$ and all sequences $A$.  Ideally, if the generalized anti-Waring number $N(k,n,r,A)$ exists, a formula for it can be derived.  We have found such a formula for some cases.  For other cases, we have computationally found and verified $N(k,n,r,A)$ with Theorem~\ref{verify_N}.

Johnson and Laughlin \cite[Theorem 1]{Johnson} proved a first result 
\begin{equation}\label{N11rN} N(1,1,r,\mathbb{N})=\sum_{i=1}^r i =  \frac{r}{2}(r+1)\end{equation}
for the case of $k=n=1$.  A similar argument is valid for general values of $n$.
 
\begin{theorem} \label{formula_N1nrN}
For positive integers $n$ and $r$, the generalized anti-Waring number is given by
$$N(1,n,r,\mathbb{N}) = \sum_{i=n}^{n+r-1}i = \frac{r}{2}(r+1)+r(n-1).$$
\end{theorem}
\begin{proof}
Clearly, the sum $\sum_{i=n}^{n+r-1}i$ is the smallest integer expressible as the sum of $r$ or more distinct integers greater than or equal to $n$. For any positive integer $x$ greater than the sum $\sum_{i=n}^{n+r-1}i$, we have 
$$x-\sum_{i=n}^{n+r-2}i &gt; n+r-1.$$
Finally, we have that 
$$x = \sum_{i=n}^{n+r-2}i+\left(x-\sum_{i=n}^{n+r-2}i\right)$$
so the integer $x$ is the sum of $r$ distinct integers greater than or equal to $n$.
\end{proof}

\begin{theorem}\label{thmN1nrst}
For positive integers $n$, $r$, and $s$ and integers $t$ such that $|t|&lt;s$ and $\gcd(s,t)=1$, the generalized anti-Waring number is given by
\begin{equation}\label{N1nrst} N(1,n,r,(si+t)_{i\in\mathbb{N}}) = 1-s+\sum_{i=n}^{n+r+s-2}(si+t).\end{equation}
\end{theorem}
Note: For the case of $s=1$ and $t=0$, this reduces to $N(1,n,r,\mathbb{N})$ and agrees with Theorem~\ref{formula_N1nrN}.
\begin{proof}
The sequence $B=(si+t)_{i\geq n}$ consists of all positive integers equivalent to $t\bmod s$ that are greater than or equal to $sn+t$.  For any positive integer $p$, the sum of any $p$ elements of $B$ is equivalent to $pt\bmod s$.  In order to express all sufficiently large integers as the sum of $r$ or more distinct elements of $B$, we need sums with the number of summands covering all equivalence classes of $\mathbb{Z}_s$.  The list $r,r+1,r+2,\ldots ,r+s-1$ contains representatives of each equivalence class in $\mathbb{Z}_s$.  Since the integers $s$ and $t$ are relatively prime, the same is true for the list $rt,(r+1)t,(r+2)t,\ldots ,(r+s-1)t$.  Hence, all sums containing between $r$ and $r+s-1$ distinct elements of $B$ will account for all sufficiently large positive integers, as we shall see.  We must determine the smallest integer not expressible by one of these sums.

For $p\in\{r,r+1,r+2,\ldots ,r+s-1\}$, let $m_p$ be the sum of the first $p$ elements of $B$, i.e.,
$$m_p=\sum_{i=n}^{n+p-1} (si+t)=s\left(\sum_{i=n}^{n+p-1} i\right)+pt.$$
As noted before, we have $m_p\equiv pt$ (mod $s$).  We also know that $m_p$ is the smallest integer equivalent to $pt\bmod s$ expressible as the sum of $r$ or more distinct elements of $B$.  Hence the integer $m_p-s$ is $(1,n,r,(si+t)_{i\in\mathbb{N}})$-bad.  If a positive integer $x\geq m_p$ is also equivalent to $pt\bmod s$, then we have $x = m_p +\ell s$ for some positive $\ell\in\mathbb{Z}$ or, equivalently,
$$x = \ell s+\sum_{i=n}^{n+p-1} (si+t)=(s(\ell+n+p-1)+t)+\sum_{i=n}^{n+p-2} (si+t).$$
Thus, all integers equivalent to $pt\bmod s$ greater than $m_p$ are expressible as the sum of $r$ or more distinct elements of $B$.  Since we have $m_p&lt;m_{p+1}$ for all $p$, the last $(1,n,r,(si+t)_{i\in\mathbb{N}})$-bad integer is $m_{r+s-1}-s$.  Therefore, the generalized anti-Waring number is $N(1,n,r,(si+t)_{i\in\mathbb{N}})=m_{r+s-1}-s+1$ which is \eqref{N1nrst}.
\end{proof}

\begin{table}[H]
\footnotesize\begin{center}
\begin{tabular}{|r|rrr|r|}
\hline
$k$ &amp; $N(k,1)$  &amp; $x$ &amp; $b$ &amp; bad count \\
\hline
1  &amp;  1&amp;1&amp;4 					&amp;0 				\\
2  &amp;  129&amp;4&amp;18 				&amp;31 				\\
3  &amp;  12759&amp;5&amp;32 			&amp;2788 			\\
4  &amp;  5134241&amp;8&amp;59 		&amp;889576 		\\
5  &amp;  67898772&amp;4&amp;45 		&amp;13912682  	\\
6  &amp;  11146309948&amp;5&amp;55	&amp;2037573096	\\
\hline
\end{tabular}
\end{center}
\caption{Values of $N(k,1,1,\mathbb{N}$)}
\label{Nk1tab}
\end{table}

\begin{table}[H]
\footnotesize\begin{center}
\begin{tabular}{|r|rrr|r|}
\hline
$k$ &amp; $N(k,1,1,\mathbb{P})$  &amp; $x$ &amp;$b$  &amp; bad count \\
\hline
1  &amp;  7&amp;6&amp;14 &amp;3\\
2  &amp;  17164&amp;54&amp;187 &amp;2438\\
3  &amp;  1866001&amp;31&amp;157 &amp;483370\\
\hline
\end{tabular}
\end{center}
\caption{Values of $N(k,1,1,\mathbb{P})$}
\label{Nk1prime}
\end{table}

For most cases, a formula for $N(k,n,r,A)$ is not known, but we can compute particular values.  In the Tables~\ref{Nk1tab} to \ref{Nk11sttabB} we list values of $N(k,n,r,A)$ along with the corresponding $x$ and $b$ that satisfy the conditions for Theorem~\ref{verify_N} hence confirming the given generalized anti-Waring number.  Tables~\ref{Nk1tab}, \ref{Nkrtab}, and \ref{Nknrtab} use $A=\mathbb{N}$.  In Table~\ref{Nk1tab} we consider $n=r=1$, i.e., the first positive integer such that it and every subsequent integer can be written as the sum $k^{\rm th}$ powers of distinct integers.  For each $k$ we also include a \emph{bad count}, i.e., the number of positive integers that cannot be written as a sum of $k^{\rm th}$ powers.  Table~\ref{Nk1prime} lists the corresponding values over the sequence of primes $\mathbb{P}$.  Table~\ref{Nkrtab} lists generalized anti-Waring numbers for fixed $n=1$ and varying $k$ and $r$.  We stopped the table at $r=36$ but were able to compute some $N(k,1,r,\mathbb{N})$ for much larger $r$.  For example, we found that $N(2,1,1000,\mathbb{N})= 333951595$ with $x=12898$ and $b=19395$.  Table~\ref{Nknrtab} lists generalized anti-Waring numbers for varying $k$, $n$, and $r$.  Tables~\ref{Nkrtab} and \ref{Nknrtab} omit generalized anti-Waring numbers when $k=1$ because a formula for $N(1,n,r,\mathbb{N})$ for all $n$ and $r$ in $\mathbb{N}$ exists by Theorem~\ref{formula_N1nrN}.  Tables~\ref{Nk11sttabA} and \ref{Nk11sttabB} list generalized anti-Waring numbers for fixed $n=1$ and $r=1$ over various sequences of the form $(si+t)_{i\in\mathbb{N}}$.	

\begin{table}[H]
\footnotesize\begin{center}
\begin{tabular}{|r|rrr|rrr|rrr|rrr|}
\hline
$r$ &amp; $N(2,r)$  &amp; $x$ &amp; $b$ &amp; $N(3,r)$ &amp; $x$ &amp; $b$ &amp; $N(4,r)$ &amp; $x$ &amp; $b$ &amp; $N(5,r)$ &amp; $x$ &amp; $b$ \\
\hline
1 &amp; 129 &amp; 4 &amp; 18 &amp; 12759 &amp; 5 &amp; 32 &amp; 5134241 &amp; 8 &amp; 59 &amp; 67898772 &amp; 4 &amp; 45 \\
2 &amp; 129 &amp; 4 &amp; 18 &amp; 12759 &amp; 5 &amp; 32 &amp; 5134241 &amp; 8 &amp; 59 &amp; 67898772 &amp; 4 &amp; 45 \\
3 &amp; 129 &amp; 4 &amp; 18 &amp; 12759 &amp; 5 &amp; 32 &amp; 5134241 &amp; 8 &amp; 59 &amp; 67898772 &amp; 4 &amp; 45 \\
4 &amp; 129 &amp; 4 &amp; 18 &amp; 12759 &amp; 5 &amp; 32 &amp; 5134241 &amp; 8 &amp; 59 &amp; 67898772 &amp; 4 &amp; 45 \\
5 &amp; 198 &amp; 6 &amp; 22 &amp; 12759 &amp; 5 &amp; 32 &amp; 5134241 &amp; 8 &amp; 59 &amp; 67898772 &amp; 4 &amp; 45 \\
6 &amp; 238 &amp; 6 &amp; 23 &amp; 15279 &amp; 6 &amp; 33 &amp; 5134241 &amp; 8 &amp; 59 &amp; 67898772 &amp; 4 &amp; 45 \\
7 &amp; 331 &amp; 8 &amp; 26 &amp; 15279 &amp; 6 &amp; 33 &amp; 5134241 &amp; 8 &amp; 59 &amp; 67898772 &amp; 4 &amp; 45 \\
8 &amp; 383 &amp; 9 &amp; 27 &amp; 15279 &amp; 6 &amp; 33 &amp; 5134241 &amp; 8 &amp; 59 &amp; 67898772 &amp; 4 &amp; 45 \\
9 &amp; 528 &amp; 10 &amp; 32 &amp; 16224 &amp; 6 &amp; 33 &amp; 5134241 &amp; 8 &amp; 59 &amp; 67898772 &amp; 4 &amp; 45 \\
10 &amp; 648 &amp; 12 &amp; 33 &amp; 18149 &amp; 6 &amp; 35 &amp; 5134241 &amp; 8 &amp; 59 &amp; 67898772 &amp; 4 &amp; 45 \\
11 &amp; 889 &amp; 14 &amp; 39 &amp; 22398 &amp; 7 &amp; 37 &amp; 5191473 &amp; 8 &amp; 59 &amp; 67898772 &amp; 4 &amp; 45 \\
12 &amp; 989 &amp; 15 &amp; 41 &amp; 24855 &amp; 7 &amp; 38 &amp; 5626194 &amp; 8 &amp; 60 &amp; 67898772 &amp; 4 &amp; 45 \\
13 &amp; 1178 &amp; 17 &amp; 44 &amp; 28887 &amp; 8 &amp; 39 &amp; 6018930 &amp; 8 &amp; 62 &amp; 71780055 &amp; 4 &amp; 46 \\
14 &amp; 1398 &amp; 19 &amp; 47 &amp; 36951 &amp; 9 &amp; 42 &amp; 6408466 &amp; 9 &amp; 62 &amp; 74729904 &amp; 4 &amp; 46 \\
15 &amp; 1723 &amp; 21 &amp; 52 &amp; 39660 &amp; 9 &amp; 43 &amp; 6664722 &amp; 9 &amp; 62 &amp; 81846431 &amp; 5 &amp; 45 \\
16 &amp; 1991 &amp; 24 &amp; 54 &amp; 49083 &amp; 10 &amp; 46 &amp; 6938867 &amp; 9 &amp; 63 &amp; 92894512 &amp; 5 &amp; 47 \\
17 &amp; 2312 &amp; 26 &amp; 58 &amp; 56076 &amp; 11 &amp; 47 &amp; 8077523 &amp; 9 &amp; 66 &amp; 95723448 &amp; 5 &amp; 47 \\
18 &amp; 2673 &amp; 28 &amp; 62 &amp; 66534 &amp; 12 &amp; 50 &amp; 8592323 &amp; 9 &amp; 67 &amp; 112031630 &amp; 5 &amp; 49 \\
19 &amp; 3048 &amp; 31 &amp; 65 &amp; 75912 &amp; 12 &amp; 52 &amp; 9269124 &amp; 10 &amp; 67 &amp; 124811198 &amp; 5 &amp; 50 \\
20 &amp; 3493 &amp; 34 &amp; 69 &amp; 87567 &amp; 13 &amp; 54 &amp; 10418260 &amp; 10 &amp; 69 &amp; 142118181 &amp; 5 &amp; 52 \\
21 &amp; 4094 &amp; 36 &amp; 75 &amp; 101093 &amp; 14 &amp; 56 &amp; 10589380 &amp; 10 &amp; 70 &amp; 163637305 &amp; 6 &amp; 52 \\
22 &amp; 4614 &amp; 39 &amp; 79 &amp; 122064 &amp; 15 &amp; 60 &amp; 12852837 &amp; 11 &amp; 72 &amp; 189572962 &amp; 6 &amp; 54 \\
23 &amp; 5139 &amp; 42 &amp; 83 &amp; 138696 &amp; 16 &amp; 62 &amp; 13199973 &amp; 11 &amp; 73 &amp; 210715205 &amp; 6 &amp; 55 \\
24 &amp; 5719 &amp; 44 &amp; 87 &amp; 156498 &amp; 17 &amp; 64 &amp; 15148358 &amp; 11 &amp; 76 &amp; 247073537 &amp; 6 &amp; 57 \\
25 &amp; 6380 &amp; 48 &amp; 91 &amp; 179520 &amp; 18 &amp; 67 &amp; 16526214 &amp; 12 &amp; 76 &amp; 285744830 &amp; 7 &amp; 57 \\
26 &amp; 7124 &amp; 51 &amp; 96 &amp; 201921 &amp; 19 &amp; 69 &amp; 17803895 &amp; 12 &amp; 78 &amp; 319712379 &amp; 7 &amp; 59 \\
27 &amp; 7953 &amp; 54 &amp; 101 &amp; 227400 &amp; 20 &amp; 72 &amp; 20499591 &amp; 13 &amp; 81 &amp; 374237223 &amp; 7 &amp; 61  \\
28 &amp; 8677 &amp; 57 &amp; 105 &amp; 256254 &amp; 22 &amp; 73 &amp; 21202776 &amp; 13 &amp; 81 &amp; 430026890 &amp; 7 &amp; 63  \\
29 &amp; 9538 &amp; 61 &amp; 109 &amp; 289869 &amp; 23 &amp; 76 &amp; 24306872 &amp; 13 &amp; 84 &amp; 491665093 &amp; 8 &amp; 64  \\
30 &amp; 10394 &amp; 63 &amp; 114 &amp; 325590 &amp; 24 &amp; 79 &amp; 25670088 &amp; 14 &amp; 84 &amp; 558015873 &amp; 8 &amp; 65 \\
31 &amp; 11559 &amp; 67 &amp; 120 &amp; 359358 &amp; 25 &amp; 82 &amp; 29819129 &amp; 14 &amp; 88 &amp; 640101337 &amp; 8 &amp; 68   \\
32 &amp; 12603 &amp; 71 &amp; 125 &amp; 401496 &amp; 26 &amp; 85 &amp; 31126025 &amp; 15 &amp; 88 &amp; 737104155 &amp; 9 &amp; 68   \\
33 &amp; 13744 &amp; 74 &amp; 130 &amp; 448503 &amp; 27 &amp; 88 &amp; 35677050 &amp; 15 &amp; 92 &amp; 839165455 &amp; 9 &amp; 71   \\
34 &amp; 14864 &amp; 78 &amp; 135 &amp; 496257 &amp; 29 &amp; 90 &amp; 38187306 &amp; 16 &amp; 92 &amp; 950792455 &amp; 9 &amp; 73   \\
35 &amp; 16253 &amp; 81 &amp; 141 &amp; 554217 &amp; 30 &amp; 93 &amp; 43256507 &amp; 16 &amp; 96 &amp; 1070200765 &amp; 10 &amp; 73   \\
36 &amp; 17529 &amp; 85 &amp; 146 &amp; 611736 &amp; 30 &amp; 97 &amp; 46180043 &amp; 17 &amp; 97 &amp; 1215652918 &amp; 10 &amp; 76   \\
\hline
\end{tabular}
\end{center}
\caption{Values of $N(k,1,r,\mathbb{N}$) and the corresponding $x$ and $b$ that satisfy Theorem~\ref{verify_N}.  Values of $N(1,n,r,\mathbb{N})$ are given by Theorem~\ref{formula_N1nrN}.}
\label{Nkrtab}
\end{table}

\begin{table}[H]
\footnotesize\begin{center}
\resizebox{.68\textwidth}{!}{%
\begin{tabular}{|r|rrr|rrr|rrr|rrr|rr}
\hline
$r$ &amp; $N(2,2,r)$ &amp; $x$ &amp; $b$ &amp; $N(3,2,r)$ &amp; $x$ &amp; $b$ &amp; $N(4,2,r)$ &amp; $x$ &amp; $b$ &amp; $N(5,2,r)$ &amp; $x$ &amp; $b$ \\
\hline
 1 &amp; 193 &amp; 5 &amp; 22 &amp; 19310 &amp; 6 &amp; 36 &amp; 6659841 &amp; 9 &amp; 62 &amp; 84038312 &amp; 5 &amp; 46 \\
 2 &amp; 193 &amp; 5 &amp; 22 &amp; 19310 &amp; 6 &amp; 36 &amp; 6659841 &amp; 9 &amp; 62 &amp; 84038312 &amp; 5 &amp; 46 \\
 3 &amp; 193 &amp; 5 &amp; 22 &amp; 19310 &amp; 6 &amp; 36 &amp; 6659841 &amp; 9 &amp; 62 &amp; 84038312 &amp; 5 &amp; 46 \\
 4 &amp; 213 &amp; 6 &amp; 22 &amp; 19310 &amp; 6 &amp; 36 &amp; 6659841 &amp; 9 &amp; 62 &amp; 84038312 &amp; 5 &amp; 46 \\
 5 &amp; 318 &amp; 7 &amp; 27 &amp; 19310 &amp; 6 &amp; 36 &amp; 6659841 &amp; 9 &amp; 62 &amp; 84038312 &amp; 5 &amp; 46 \\
 6 &amp; 334 &amp; 8 &amp; 26 &amp; 19310 &amp; 6 &amp; 36 &amp; 6692881 &amp; 9 &amp; 62 &amp; 84038312 &amp; 5 &amp; 46 \\
 7 &amp; 398 &amp; 9 &amp; 27 &amp; 19310 &amp; 6 &amp; 36 &amp; 6692881 &amp; 9 &amp; 62 &amp; 84038312 &amp; 5 &amp; 46 \\
 8 &amp; 527 &amp; 10 &amp; 32 &amp; 19310 &amp; 6 &amp; 36 &amp; 6692881 &amp; 9 &amp; 62 &amp; 84038312 &amp; 5 &amp; 46 \\
 9 &amp; 647 &amp; 12 &amp; 33 &amp; 20885 &amp; 7 &amp; 36 &amp; 6778897 &amp; 9 &amp; 62 &amp; 84038312 &amp; 5 &amp; 46 \\
 10 &amp; 888 &amp; 14 &amp; 39 &amp; 24098 &amp; 7 &amp; 38 &amp; 6778897 &amp; 9 &amp; 62 &amp; 84038312 &amp; 5 &amp; 46 \\
\hline\hline
$r$ &amp; $N(2,3,r)$ &amp; $x$ &amp; $b$ &amp; $N(3,3,r)$ &amp; $x$ &amp; $b$ &amp; $N(4,3,r)$ &amp; $x$ &amp; $b$ &amp; $N(5,3,r)$ &amp; $x$ &amp; $b$ \\
\hline
1 &amp; 224 &amp; 6 &amp; 23 &amp; 23775 &amp; 7 &amp; 38 &amp; 7076321 &amp; 9 &amp; 63 &amp; 110100822 &amp; 5 &amp; 49 \\
2 &amp; 224 &amp; 6 &amp; 23 &amp; 23775 &amp; 7 &amp; 38 &amp; 7076321 &amp; 9 &amp; 63 &amp; 110100822 &amp; 5 &amp; 49 \\
3 &amp; 233 &amp; 6 &amp; 23 &amp; 23775 &amp; 7 &amp; 38 &amp; 7076321 &amp; 9 &amp; 63 &amp; 110100822 &amp; 5 &amp; 49 \\
4 &amp; 314 &amp; 7 &amp; 26 &amp; 23775 &amp; 7 &amp; 38 &amp; 7076321 &amp; 9 &amp; 63 &amp; 110100822 &amp; 5 &amp; 49 \\
5 &amp; 330 &amp; 8 &amp; 26 &amp; 23775 &amp; 7 &amp; 38 &amp; 7076321 &amp; 9 &amp; 63 &amp; 110100822 &amp; 5 &amp; 49 \\
6 &amp; 418 &amp; 9 &amp; 28 &amp; 23775 &amp; 7 &amp; 38 &amp; 7076321 &amp; 9 &amp; 63 &amp; 110100822 &amp; 5 &amp; 49 \\
7 &amp; 523 &amp; 10 &amp; 32 &amp; 23775 &amp; 7 &amp; 38 &amp; 7103505 &amp; 9 &amp; 63 &amp; 110100822 &amp; 5 &amp; 49 \\
8 &amp; 643 &amp; 12 &amp; 33 &amp; 24756 &amp; 7 &amp; 38 &amp; 7103505 &amp; 9 &amp; 63 &amp; 110100822 &amp; 5 &amp; 49 \\
9 &amp; 884 &amp; 14 &amp; 39 &amp; 28221 &amp; 7 &amp; 41 &amp; 7103505 &amp; 9 &amp; 63 &amp; 110100822 &amp; 5 &amp; 49 \\
10 &amp; 984 &amp; 15 &amp; 41 &amp; 28950 &amp; 8 &amp; 40 &amp; 7103505 &amp; 9 &amp; 63 &amp; 110100822 &amp; 5 &amp; 49 \\
\hline\hline
$r$ &amp; $N(2,4,r)$ &amp; $x$ &amp; $b$ &amp; $N(3,4,r)$ &amp; $x$ &amp; $b$ &amp; $N(4,4,r)$ &amp; $x$ &amp; $b$ &amp; $N(5,4,r)$ &amp; $x$ &amp; $b$ \\
\hline
1 &amp; 385 &amp; 8 &amp; 30 &amp; 26862 &amp; 7 &amp; 40 &amp; 8912545 &amp; 9 &amp; 68 &amp; 129436797 &amp; 5 &amp; 51 \\
2 &amp; 385 &amp; 8 &amp; 30 &amp; 26862 &amp; 7 &amp; 40 &amp; 8912545 &amp; 9 &amp; 68 &amp; 129436797 &amp; 5 &amp; 51 \\
3 &amp; 385 &amp; 8 &amp; 29 &amp; 26862 &amp; 7 &amp; 40 &amp; 8912545 &amp; 9 &amp; 68 &amp; 129436797 &amp; 5 &amp; 51 \\
4 &amp; 385 &amp; 8 &amp; 28 &amp; 26862 &amp; 7 &amp; 40 &amp; 8912545 &amp; 9 &amp; 68 &amp; 129436797 &amp; 5 &amp; 51 \\
5 &amp; 453 &amp; 9 &amp; 30 &amp; 26862 &amp; 7 &amp; 40 &amp; 8912545 &amp; 9 &amp; 68 &amp; 129436797 &amp; 5 &amp; 51 \\
6 &amp; 558 &amp; 10 &amp; 33 &amp; 26862 &amp; 7 &amp; 40 &amp; 8912545 &amp; 9 &amp; 68 &amp; 129436797 &amp; 5 &amp; 51 \\
7 &amp; 634 &amp; 12 &amp; 34 &amp; 27528 &amp; 7 &amp; 40 &amp; 8912545 &amp; 9 &amp; 68 &amp; 129436797 &amp; 5 &amp; 51 \\
8 &amp; 875 &amp; 14 &amp; 39 &amp; 28194 &amp; 7 &amp; 41 &amp; 8912545 &amp; 9 &amp; 68 &amp; 129436797 &amp; 5 &amp; 51 \\
9 &amp; 999 &amp; 15 &amp; 41 &amp; 30111 &amp; 8 &amp; 40 &amp; 8912545 &amp; 9 &amp; 68 &amp; 129436797 &amp; 5 &amp; 51 \\
10 &amp; 1164 &amp; 17 &amp; 43 &amp; 33234 &amp; 8 &amp; 42 &amp; 8912545 &amp; 9 &amp; 68 &amp; 130964972 &amp; 5 &amp; 51 \\
\hline\hline
$r$ &amp; $N(2,5,r)$ &amp; $x$ &amp; $b$ &amp; $N(3,5,r)$ &amp; $x$ &amp; $b$ &amp; $N(4,5,r)$ &amp; $x$ &amp; $b$ &amp; $N(5,5,r)$ &amp; $x$ &amp; $b$ \\
\hline
1 &amp; 493 &amp; 9 &amp; 34 &amp; 34844 &amp; 8 &amp; 43 &amp; 9292705 &amp; 9 &amp; 69 &amp; 167956256 &amp; 5 &amp; 54 \\
2 &amp; 493 &amp; 9 &amp; 33 &amp; 34844 &amp; 8 &amp; 43 &amp; 9292705 &amp; 9 &amp; 69 &amp; 167956256 &amp; 5 &amp; 54 \\
3 &amp; 493 &amp; 9 &amp; 32 &amp; 34844 &amp; 8 &amp; 43 &amp; 9292705 &amp; 9 &amp; 69 &amp; 167956256 &amp; 5 &amp; 54 \\
4 &amp; 494 &amp; 9 &amp; 32 &amp; 34844 &amp; 8 &amp; 43 &amp; 9292705 &amp; 9 &amp; 69 &amp; 167956256 &amp; 5 &amp; 54 \\
5 &amp; 542 &amp; 10 &amp; 33 &amp; 34844 &amp; 8 &amp; 43 &amp; 9292705 &amp; 9 &amp; 69 &amp; 167956256 &amp; 5 &amp; 54 \\
6 &amp; 670 &amp; 12 &amp; 35 &amp; 34844 &amp; 8 &amp; 43 &amp; 9377041 &amp; 9 &amp; 69 &amp; 167956256 &amp; 5 &amp; 54 \\
7 &amp; 883 &amp; 14 &amp; 39 &amp; 35060 &amp; 8 &amp; 43 &amp; 9377041 &amp; 9 &amp; 69 &amp; 167956256 &amp; 5 &amp; 54 \\
8 &amp; 983 &amp; 15 &amp; 41 &amp; 35060 &amp; 8 &amp; 43 &amp; 9377041 &amp; 10 &amp; 68 &amp; 167956256 &amp; 5 &amp; 54 \\
9 &amp; 1188 &amp; 17 &amp; 44 &amp; 38048 &amp; 8 &amp; 44 &amp; 9377041 &amp; 10 &amp; 68 &amp; 167956256 &amp; 6 &amp; 53 \\
10 &amp; 1412 &amp; 19 &amp; 47 &amp; 43880 &amp; 9 &amp; 45 &amp; 9377041 &amp; 10 &amp; 68 &amp; 167956256 &amp; 5 &amp; 54 \\
\hline\hline
$r$ &amp; $N(2,6,r)$ &amp; $x$ &amp; $b$ &amp; $N(3,6,r)$ &amp; $x$ &amp; $b$ &amp; $N(4,6,r)$ &amp; $x$ &amp; $b$ &amp; $N(5,6,r)$ &amp; $x$ &amp; $b$ \\
\hline
1 &amp; 637 &amp; 10 &amp; 37 &amp; 40416 &amp; 8 &amp; 45 &amp; 11728881 &amp; 10 &amp; 72 &amp; 191116579 &amp; 6 &amp; 54 \\
2 &amp; 637 &amp; 10 &amp; 37 &amp; 40416 &amp; 8 &amp; 45 &amp; 11728881 &amp; 10 &amp; 72 &amp; 191116579 &amp; 6 &amp; 54 \\
3 &amp; 637 &amp; 10 &amp; 37 &amp; 40416 &amp; 8 &amp; 45 &amp; 11728881 &amp; 10 &amp; 72 &amp; 191116579 &amp; 6 &amp; 54 \\
4 &amp; 637 &amp; 11 &amp; 35 &amp; 40416 &amp; 8 &amp; 45 &amp; 11728881 &amp; 10 &amp; 72 &amp; 191116579 &amp; 6 &amp; 54 \\
5 &amp; 834 &amp; 13 &amp; 40 &amp; 40416 &amp; 8 &amp; 45 &amp; 11728881 &amp; 10 &amp; 72 &amp; 191116579 &amp; 6 &amp; 54 \\
6 &amp; 870 &amp; 13 &amp; 40 &amp; 40416 &amp; 8 &amp; 45 &amp; 11728881 &amp; 10 &amp; 72 &amp; 191116579 &amp; 6 &amp; 54 \\
7 &amp; 958 &amp; 15 &amp; 40 &amp; 40416 &amp; 8 &amp; 45 &amp; 11728881 &amp; 10 &amp; 72 &amp; 191116579 &amp; 6 &amp; 54 \\
8 &amp; 1163 &amp; 17 &amp; 43 &amp; 41450 &amp; 9 &amp; 44 &amp; 11728881 &amp; 10 &amp; 72 &amp; 191116579 &amp; 6 &amp; 54 \\
9 &amp; 1387 &amp; 19 &amp; 46 &amp; 48066 &amp; 9 &amp; 47 &amp; 11728881 &amp; 10 &amp; 72 &amp; 191116579 &amp; 6 &amp; 54 \\
10 &amp; 1668 &amp; 21 &amp; 51 &amp; 49893 &amp; 10 &amp; 46 &amp; 11728881 &amp; 10 &amp; 72 &amp; 191116579 &amp; 6 &amp; 54 \\
\hline
\end{tabular}
}
\end{center}
\caption{Values of $N(k,n,r,\mathbb{N}$) for $n&gt;1$ and the corresponding $x$ and $b$ that satisfy Theorem~\ref{verify_N}.   Values of $N(1,n,r,\mathbb{N})$ are given by Theorem~\ref{formula_N1nrN}.}
\label{Nknrtab}
\end{table}

\begin{table}[H]
\footnotesize
\begin{center}
\resizebox{.40\textwidth}{!}{%
\begin{tabular}{|r|rrr|rrr|}
\hline
$(s,t)$ &amp; $k=2$  &amp; $x$ &amp; $b$ &amp; $k=3$ &amp; $x$ &amp; $b$ \\ \hline
$(2,-1)$&amp;	1923 &amp; 18&amp;64 &amp; 			212595 &amp; 15&amp;77 \\
$(2,+1)$&amp;2355 &amp;20 &amp;71 &amp; 			266459 &amp; 16&amp;83  \\ \hline
$(3,-2)$&amp;	3250 &amp; 23&amp; 83&amp; 			942316 &amp; 25&amp;126 \\
$(3,-1)$&amp;	3014 &amp;22 &amp;80 &amp; 			957226 &amp; 25&amp;126 \\
$(3,+1)$&amp;4093 &amp; 26&amp;92 &amp; 			1103569 &amp; 26&amp;132\\
$(3,+2)$&amp;4414 &amp; 27&amp;96 &amp; 			1181758 &amp; 27&amp;135\\ \hline
$(4,-3)$&amp;	10588 &amp; 42&amp;148 &amp; 			2576040 &amp; 35&amp;174\\
$(4,-1)$&amp;	11268 &amp; 43&amp;153 &amp; 			3026615 &amp; 37&amp;184\\
$(4,+1)$&amp;13708 &amp; 48&amp;167 &amp; 			3152462 &amp; 37&amp;187 \\
$(4,+3)$&amp;14948 &amp; 50&amp;175 &amp; 			3534459 &amp; 39&amp;193\\ \hline
$(5,-4)$&amp;	14900 &amp; 50&amp;174 &amp; 			6146241 &amp; 47&amp;232\\
$(5,-3)$&amp;	14121 &amp; 49&amp;170 &amp; 			6373428 &amp; 47&amp;236\\
$(5,-2)$&amp;	16810 &amp; 53&amp;186 &amp; 			6672804 &amp; 48&amp;239\\
$(5,-1)$&amp;	16379 &amp; 52&amp;184 &amp; 			7077048 &amp; 49&amp;244\\
$(5,+1)$&amp;17242 &amp; 54&amp;187 &amp; 			7165274 &amp; 49&amp;245\\
$(5,+2)$&amp;19090 &amp; 57&amp;198 &amp; 			7526193 &amp; 50&amp;249\\
$(5,+3)$&amp;19690 &amp; 58&amp;201 &amp; 			7821959 &amp; 51&amp;252\\
$(5,+4)$&amp;19799 &amp; 58&amp;201 &amp; 			8326652 &amp; 52&amp;257\\ \hline
$(6,-5)$&amp;	255964 &amp; 209&amp;717 &amp; 		32025571 &amp; 82&amp;402\\
$(6,-1)$&amp;	261868 &amp; 211&amp;727 &amp; 		35431051 &amp; 85&amp;416\\
$(6,+1)$&amp;270796 &amp; 215&amp;738 &amp; 		38008681 &amp; 87&amp;426 \\
$(6,+5)$&amp;282028 &amp; 219&amp;754 &amp; 		40622251 &amp; 88&amp;436 \\ \hline
$(7,-6)$&amp;	44329 &amp; 87&amp;300 &amp; 			24233667 &amp; 74&amp;367 \\
$(7,-5)$&amp;	45769 &amp; 88&amp;305 &amp; 			23668124 &amp; 74&amp;363 \\
$(7,-4)$&amp;	49737 &amp; 92&amp;317 &amp; 			25473560 &amp; 76&amp;373 \\
$(7,-3)$&amp;	49009 &amp; 91&amp;315 &amp; 			26139255 &amp; 76&amp;376 \\
$(7,-2)$&amp;	48989 &amp; 91&amp;315 &amp; 			27035708 &amp; 77&amp;380 \\
$(7,-1)$&amp;	49537 &amp; 92&amp;317 &amp; 			27348027 &amp; 77&amp;382 \\
$(7,+1)$&amp;51889 &amp; 94&amp;324 &amp; 			28963994 &amp; 79&amp;389 \\
$(7,+2)$&amp;55884 &amp; 97&amp;337 &amp; 			28297320 &amp; 78&amp;387 \\
$(7,+3)$&amp;54217 &amp; 96&amp;331 &amp; 			30183369 &amp; 80&amp;394  \\
$(7,+4)$&amp;60377 &amp; 101&amp;350 &amp; 		28992218 &amp; 79&amp;389 \\ 
$(7,+5)$&amp;58292 &amp; 99&amp;344 &amp; 			31374203 &amp; 81&amp;400  \\
$(7,+6)$&amp;63453 &amp; 104&amp;358 &amp; 		31015095 &amp; 81&amp;397 \\ \hline
$(8,-7)$&amp;	183828 &amp; 177&amp;608 &amp; 		43603746 &amp; 91&amp;445  \\
$(8,-5)$&amp;	186684 &amp; 178&amp;614 &amp; 		44323025 &amp; 91&amp;448 \\
$(8,-3)$&amp;	192748 &amp; 181&amp;623 &amp; 		44594177 &amp; 91&amp;449  \\
$(8,-1)$&amp;	199124 &amp; 184&amp;634 &amp; 		49916598 &amp; 95&amp;466  \\
$(8,+1)$&amp;208164 &amp; 188&amp;648 &amp; 		51794250 &amp; 96&amp;472  \\
$(8,+3)$&amp;216940 &amp; 192&amp;661 &amp; 		53940372 &amp; 97&amp;479  \\
$(8,+5)$&amp;223884 &amp; 195&amp;672 &amp; 		53774817 &amp; 97&amp;478 \\
$(8,+7)$&amp;227204 &amp; 197&amp;676 &amp; 		55157135 &amp; 98&amp;482 \\ \hline
$(9,-8)$&amp;	104873 &amp; 134&amp;460 &amp; 		316621582 &amp; 176&amp;861  \\
$(9,-7)$&amp;	114857 &amp; 140&amp;481 &amp; 		317215246 &amp; 176&amp;862  \\
$(9,-5)$&amp;	114653 &amp; 140&amp;481 &amp; 		327375655 &amp; 178&amp;871  \\
$(9,-4)$&amp;	118829 &amp; 142&amp;490 &amp; 		329700964 &amp; 179&amp;872  \\
$(9,-2)$&amp;	120113 &amp; 143&amp;492 &amp; 		338139583 &amp; 180&amp;880  \\
$(9,-1)$&amp;	130217 &amp; 149&amp;512 &amp; 		339498184 &amp; 180&amp;882  \\
$(9,+1)$&amp;134681 &amp; 151&amp;522 &amp; 		352115215 &amp; 183&amp;891 \\
$(9,+2)$&amp;129149 &amp; 148&amp;511 &amp; 		358747834 &amp; 184&amp;897 \\
$(9,+4)$&amp;137873 &amp; 153&amp;528 &amp; 		371854375 &amp; 186&amp;908 \\
$(9,+5)$&amp;141329 &amp; 155&amp;534 &amp; 		365220868 &amp; 185&amp;902 \\
$(9,+7)$&amp;142825 &amp; 156&amp;536 &amp; 		383482411 &amp; 188&amp;917 \\
$(9,+8)$&amp;149990 &amp; 160&amp;549 &amp; 		376489804 &amp; 187&amp;911  \\ 
\hline
\end{tabular}
}
\end{center}
\caption{Values of $N(k,1,1,(si+t)_{i\in\mathbb{N}})$ and the corresponding $x$ and $b$ that satisfy Theorem~\ref{verify_N}.  The generalized anti-Waring number $N(k,n,r,(si+t)_{i\in\mathbb{N}})$ does not exist if $\gcd (s,t) &gt; 1$ by Theorem~\ref{dne_gcd}, and values of $N(1,n,r,(si+t)_{i\in\mathbb{N}})$ are given by Theorem~\ref{thmN1nrst}.}
\label{Nk11sttabA}
\end{table}

\begin{table}[H]
\footnotesize\begin{center}
\begin{tabular}{|r|rrr|rrr|rr|r|rrr|}
\hline
$(s,t)$ &amp; $k=2$  &amp; $x$ &amp; $b$ &amp; $k=3$ &amp; $x$ &amp; $b$ \\ \hline
$(10,-1)$&amp;2866844&amp;701&amp;2396&amp;  167900541&amp;142&amp;698 \\
$(10,+1)$&amp;2770803&amp;689&amp;2356&amp;  164930981&amp;142&amp;693 \\ 
$(11,-1)$&amp;251377&amp;207&amp;711&amp;  188148921&amp;148&amp;724 \\
$(11,+1)$&amp;260001&amp;211&amp;723&amp;  200560127&amp;151&amp;740 \\
$(12,-1)$&amp;1186948&amp;451&amp;1543&amp;1871937463&amp;320&amp;1555 \\
$(12,+1)$&amp;1207948&amp;455&amp;1556&amp;1897625923&amp;321&amp;1562 \\ 
 $(13,-1)$&amp;484333&amp;288&amp;986&amp;427144568&amp;195&amp;951 \\
$(13,+1)$&amp;498269&amp;292&amp;1000&amp;434996727&amp;196&amp;957 \\
$(14,-1)$&amp;14209388&amp;1561&amp;5333&amp;718660158&amp;232&amp;1130 \\
$(14,+1)$&amp;14254244&amp;1563&amp;5342&amp;750996509&amp;235&amp;1148 \\ 
 $(15,-1)$&amp;878885&amp;388&amp;1328&amp;7192487965&amp;501&amp;2434 \\
$(15,+1)$&amp;890945&amp;390&amp;1338&amp;7247153841&amp;502&amp;2440 \\
 $(16,-1)$&amp;4345668&amp;863&amp;2950&amp;1162662009&amp;272&amp;1328 \\
 $(16,+1)$&amp;4411364&amp;869&amp;2973&amp;1188105593&amp;274&amp;1337 \\
 $(17,-1)$&amp;1468737&amp;501&amp;1717&amp;1528625985&amp;298&amp;1454 \\
$(17,+1)$&amp;1487777&amp;505&amp;1727&amp;1574453445&amp;302&amp;1468 \\
$(18,-1)$&amp;47752420&amp;2862&amp;9774&amp;23390399911&amp;742&amp;3606 \\
$(18,+1)$&amp;47891524&amp;2866&amp;9789&amp;23431535880&amp;743&amp;3607 \\ 
$(19,-1)$&amp;2296953&amp;627&amp;2146&amp;2670453204&amp;360&amp;1750 \\
$(19,+1)$&amp;2330393&amp;632&amp;2161&amp;2654207231&amp;359&amp;1746 \\
$(20,-1)$&amp;12065164&amp;1438&amp;4915&amp;3392160594&amp;390&amp;1895 \\
$(20,+1)$&amp;12241324&amp;1449&amp;4950&amp;3426870488&amp;391&amp;1901 \\ 
\hline
\end{tabular}
\end{center}
\caption{Additional values of $N(k,1,1,(si+t)_{i\in\mathbb{N}})$ and the corresponding $x$ and $b$ that satisfy Theorem~\ref{verify_N}.  The generalized anti-Waring number $N(k,n,r,(si+t)_{i\in\mathbb{N}})$ does not exist if $\gcd (s,t) &gt; 1$ by Theorem~\ref{dne_gcd}, and values of $N(1,n,r,(si+t)_{i\in\mathbb{N}})$ are given by Theorem~\ref{thmN1nrst}.}
\label{Nk11sttabB}
\end{table}

\section{Future work}

With enough time and computing power, we can compute any values of $N(k,n,r,A)$ that exist.  However, we have only found a formula for cases with $k=1$.

Some simple inequalities involving $N(k,n,r,A)$ are clear.  For example, for $i\leq j$ we have the inequalities $N(k,i,r,A)\leq N(k,j,r,A)$ and $N(k,n,i,A)\leq N(k,n,j,A)$ when each exists.  We are unable to prove the inequality $N(k,n,r,A)\leq N(k+1,n,r,A)$ even though all data seem to emphatically support it.

We have found and considered several algorithms for generating good numbers.  However, none reveal a formula for the largest bad number, i.e., threshold of completeness for $k&gt;1$.

\section{Acknowledgments}

We thank the editor and the anonymous referee for their time and consideration.  The referee's report was thorough and included valuable suggestions.

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\noindent 2010 {\it Mathematics Subject Classification}:  Primary
11P05; Secondary 05A17.

\noindent \emph{Keywords: } complete sequence,
sum of powers, anti-Waring number.

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\noindent (Concerned with sequence
\seqnum{A001661}.)

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\hrule
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\vspace*{+.1in}
\noindent
Received June 18 2015;
revised versions received  September 13 2015; September 21 2015.
Published in {\it Journal of Integer Sequences}, September 24 2015.

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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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\end{document}

                                                                                

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