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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}

\vskip .2 in


\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 $d>1$, 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 $d>1$, 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 $a_i\in A$,
$$a_i^k \bmod 6\equiv \begin{cases} 1,&\text{if $i=1$;} \\ 2^k \bmod 6,&\text{if $i=2$;} \\0, &\text{if $i>2$.}\end{cases}$$
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 $\alpha_n>0$, 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 $0<a_i\leq a_{i+1}$ 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} $\displaystyle 0<\left(\sum_{i=n}^{n+r-2}a_i^k\right)+2(m-x)^k-(m+1)^k$ 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 $m-x<a_n$, 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 
$$x-\sum_{i=n}^{n+r-2}i > n+r-1.$$
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 $|t|<s$ 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 $m_p<m_{p+1}$ 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
$k$ & $N(k,1)$  & $x$ & $b$ & bad count \\
\hline
1  &  1&1&4 					&0 				\\
2  &  129&4&18 				&31 				\\
3  &  12759&5&32 			&2788 			\\
4  &  5134241&8&59 		&889576 		\\
5  &  67898772&4&45 		&13912682  	\\
6  &  11146309948&5&55	&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
$k$ & $N(k,1,1,\mathbb{P})$  & $x$ &$b$  & bad count \\
\hline
1  &  7&6&14 &3\\
2  &  17164&54&187 &2438\\
3  &  1866001&31&157 &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
$r$ & $N(2,r)$  & $x$ & $b$ & $N(3,r)$ & $x$ & $b$ & $N(4,r)$ & $x$ & $b$ & $N(5,r)$ & $x$ & $b$ \\
\hline
1 & 129 & 4 & 18 & 12759 & 5 & 32 & 5134241 & 8 & 59 & 67898772 & 4 & 45 \\
2 & 129 & 4 & 18 & 12759 & 5 & 32 & 5134241 & 8 & 59 & 67898772 & 4 & 45 \\
3 & 129 & 4 & 18 & 12759 & 5 & 32 & 5134241 & 8 & 59 & 67898772 & 4 & 45 \\
4 & 129 & 4 & 18 & 12759 & 5 & 32 & 5134241 & 8 & 59 & 67898772 & 4 & 45 \\
5 & 198 & 6 & 22 & 12759 & 5 & 32 & 5134241 & 8 & 59 & 67898772 & 4 & 45 \\
6 & 238 & 6 & 23 & 15279 & 6 & 33 & 5134241 & 8 & 59 & 67898772 & 4 & 45 \\
7 & 331 & 8 & 26 & 15279 & 6 & 33 & 5134241 & 8 & 59 & 67898772 & 4 & 45 \\
8 & 383 & 9 & 27 & 15279 & 6 & 33 & 5134241 & 8 & 59 & 67898772 & 4 & 45 \\
9 & 528 & 10 & 32 & 16224 & 6 & 33 & 5134241 & 8 & 59 & 67898772 & 4 & 45 \\
10 & 648 & 12 & 33 & 18149 & 6 & 35 & 5134241 & 8 & 59 & 67898772 & 4 & 45 \\
11 & 889 & 14 & 39 & 22398 & 7 & 37 & 5191473 & 8 & 59 & 67898772 & 4 & 45 \\
12 & 989 & 15 & 41 & 24855 & 7 & 38 & 5626194 & 8 & 60 & 67898772 & 4 & 45 \\
13 & 1178 & 17 & 44 & 28887 & 8 & 39 & 6018930 & 8 & 62 & 71780055 & 4 & 46 \\
14 & 1398 & 19 & 47 & 36951 & 9 & 42 & 6408466 & 9 & 62 & 74729904 & 4 & 46 \\
15 & 1723 & 21 & 52 & 39660 & 9 & 43 & 6664722 & 9 & 62 & 81846431 & 5 & 45 \\
16 & 1991 & 24 & 54 & 49083 & 10 & 46 & 6938867 & 9 & 63 & 92894512 & 5 & 47 \\
17 & 2312 & 26 & 58 & 56076 & 11 & 47 & 8077523 & 9 & 66 & 95723448 & 5 & 47 \\
18 & 2673 & 28 & 62 & 66534 & 12 & 50 & 8592323 & 9 & 67 & 112031630 & 5 & 49 \\
19 & 3048 & 31 & 65 & 75912 & 12 & 52 & 9269124 & 10 & 67 & 124811198 & 5 & 50 \\
20 & 3493 & 34 & 69 & 87567 & 13 & 54 & 10418260 & 10 & 69 & 142118181 & 5 & 52 \\
21 & 4094 & 36 & 75 & 101093 & 14 & 56 & 10589380 & 10 & 70 & 163637305 & 6 & 52 \\
22 & 4614 & 39 & 79 & 122064 & 15 & 60 & 12852837 & 11 & 72 & 189572962 & 6 & 54 \\
23 & 5139 & 42 & 83 & 138696 & 16 & 62 & 13199973 & 11 & 73 & 210715205 & 6 & 55 \\
24 & 5719 & 44 & 87 & 156498 & 17 & 64 & 15148358 & 11 & 76 & 247073537 & 6 & 57 \\
25 & 6380 & 48 & 91 & 179520 & 18 & 67 & 16526214 & 12 & 76 & 285744830 & 7 & 57 \\
26 & 7124 & 51 & 96 & 201921 & 19 & 69 & 17803895 & 12 & 78 & 319712379 & 7 & 59 \\
27 & 7953 & 54 & 101 & 227400 & 20 & 72 & 20499591 & 13 & 81 & 374237223 & 7 & 61  \\
28 & 8677 & 57 & 105 & 256254 & 22 & 73 & 21202776 & 13 & 81 & 430026890 & 7 & 63  \\
29 & 9538 & 61 & 109 & 289869 & 23 & 76 & 24306872 & 13 & 84 & 491665093 & 8 & 64  \\
30 & 10394 & 63 & 114 & 325590 & 24 & 79 & 25670088 & 14 & 84 & 558015873 & 8 & 65 \\
31 & 11559 & 67 & 120 & 359358 & 25 & 82 & 29819129 & 14 & 88 & 640101337 & 8 & 68   \\
32 & 12603 & 71 & 125 & 401496 & 26 & 85 & 31126025 & 15 & 88 & 737104155 & 9 & 68   \\
33 & 13744 & 74 & 130 & 448503 & 27 & 88 & 35677050 & 15 & 92 & 839165455 & 9 & 71   \\
34 & 14864 & 78 & 135 & 496257 & 29 & 90 & 38187306 & 16 & 92 & 950792455 & 9 & 73   \\
35 & 16253 & 81 & 141 & 554217 & 30 & 93 & 43256507 & 16 & 96 & 1070200765 & 10 & 73   \\
36 & 17529 & 85 & 146 & 611736 & 30 & 97 & 46180043 & 17 & 97 & 1215652918 & 10 & 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
$r$ & $N(2,2,r)$ & $x$ & $b$ & $N(3,2,r)$ & $x$ & $b$ & $N(4,2,r)$ & $x$ & $b$ & $N(5,2,r)$ & $x$ & $b$ \\
\hline
 1 & 193 & 5 & 22 & 19310 & 6 & 36 & 6659841 & 9 & 62 & 84038312 & 5 & 46 \\
 2 & 193 & 5 & 22 & 19310 & 6 & 36 & 6659841 & 9 & 62 & 84038312 & 5 & 46 \\
 3 & 193 & 5 & 22 & 19310 & 6 & 36 & 6659841 & 9 & 62 & 84038312 & 5 & 46 \\
 4 & 213 & 6 & 22 & 19310 & 6 & 36 & 6659841 & 9 & 62 & 84038312 & 5 & 46 \\
 5 & 318 & 7 & 27 & 19310 & 6 & 36 & 6659841 & 9 & 62 & 84038312 & 5 & 46 \\
 6 & 334 & 8 & 26 & 19310 & 6 & 36 & 6692881 & 9 & 62 & 84038312 & 5 & 46 \\
 7 & 398 & 9 & 27 & 19310 & 6 & 36 & 6692881 & 9 & 62 & 84038312 & 5 & 46 \\
 8 & 527 & 10 & 32 & 19310 & 6 & 36 & 6692881 & 9 & 62 & 84038312 & 5 & 46 \\
 9 & 647 & 12 & 33 & 20885 & 7 & 36 & 6778897 & 9 & 62 & 84038312 & 5 & 46 \\
 10 & 888 & 14 & 39 & 24098 & 7 & 38 & 6778897 & 9 & 62 & 84038312 & 5 & 46 \\
\hline\hline
$r$ & $N(2,3,r)$ & $x$ & $b$ & $N(3,3,r)$ & $x$ & $b$ & $N(4,3,r)$ & $x$ & $b$ & $N(5,3,r)$ & $x$ & $b$ \\
\hline
1 & 224 & 6 & 23 & 23775 & 7 & 38 & 7076321 & 9 & 63 & 110100822 & 5 & 49 \\
2 & 224 & 6 & 23 & 23775 & 7 & 38 & 7076321 & 9 & 63 & 110100822 & 5 & 49 \\
3 & 233 & 6 & 23 & 23775 & 7 & 38 & 7076321 & 9 & 63 & 110100822 & 5 & 49 \\
4 & 314 & 7 & 26 & 23775 & 7 & 38 & 7076321 & 9 & 63 & 110100822 & 5 & 49 \\
5 & 330 & 8 & 26 & 23775 & 7 & 38 & 7076321 & 9 & 63 & 110100822 & 5 & 49 \\
6 & 418 & 9 & 28 & 23775 & 7 & 38 & 7076321 & 9 & 63 & 110100822 & 5 & 49 \\
7 & 523 & 10 & 32 & 23775 & 7 & 38 & 7103505 & 9 & 63 & 110100822 & 5 & 49 \\
8 & 643 & 12 & 33 & 24756 & 7 & 38 & 7103505 & 9 & 63 & 110100822 & 5 & 49 \\
9 & 884 & 14 & 39 & 28221 & 7 & 41 & 7103505 & 9 & 63 & 110100822 & 5 & 49 \\
10 & 984 & 15 & 41 & 28950 & 8 & 40 & 7103505 & 9 & 63 & 110100822 & 5 & 49 \\
\hline\hline
$r$ & $N(2,4,r)$ & $x$ & $b$ & $N(3,4,r)$ & $x$ & $b$ & $N(4,4,r)$ & $x$ & $b$ & $N(5,4,r)$ & $x$ & $b$ \\
\hline
1 & 385 & 8 & 30 & 26862 & 7 & 40 & 8912545 & 9 & 68 & 129436797 & 5 & 51 \\
2 & 385 & 8 & 30 & 26862 & 7 & 40 & 8912545 & 9 & 68 & 129436797 & 5 & 51 \\
3 & 385 & 8 & 29 & 26862 & 7 & 40 & 8912545 & 9 & 68 & 129436797 & 5 & 51 \\
4 & 385 & 8 & 28 & 26862 & 7 & 40 & 8912545 & 9 & 68 & 129436797 & 5 & 51 \\
5 & 453 & 9 & 30 & 26862 & 7 & 40 & 8912545 & 9 & 68 & 129436797 & 5 & 51 \\
6 & 558 & 10 & 33 & 26862 & 7 & 40 & 8912545 & 9 & 68 & 129436797 & 5 & 51 \\
7 & 634 & 12 & 34 & 27528 & 7 & 40 & 8912545 & 9 & 68 & 129436797 & 5 & 51 \\
8 & 875 & 14 & 39 & 28194 & 7 & 41 & 8912545 & 9 & 68 & 129436797 & 5 & 51 \\
9 & 999 & 15 & 41 & 30111 & 8 & 40 & 8912545 & 9 & 68 & 129436797 & 5 & 51 \\
10 & 1164 & 17 & 43 & 33234 & 8 & 42 & 8912545 & 9 & 68 & 130964972 & 5 & 51 \\
\hline\hline
$r$ & $N(2,5,r)$ & $x$ & $b$ & $N(3,5,r)$ & $x$ & $b$ & $N(4,5,r)$ & $x$ & $b$ & $N(5,5,r)$ & $x$ & $b$ \\
\hline
1 & 493 & 9 & 34 & 34844 & 8 & 43 & 9292705 & 9 & 69 & 167956256 & 5 & 54 \\
2 & 493 & 9 & 33 & 34844 & 8 & 43 & 9292705 & 9 & 69 & 167956256 & 5 & 54 \\
3 & 493 & 9 & 32 & 34844 & 8 & 43 & 9292705 & 9 & 69 & 167956256 & 5 & 54 \\
4 & 494 & 9 & 32 & 34844 & 8 & 43 & 9292705 & 9 & 69 & 167956256 & 5 & 54 \\
5 & 542 & 10 & 33 & 34844 & 8 & 43 & 9292705 & 9 & 69 & 167956256 & 5 & 54 \\
6 & 670 & 12 & 35 & 34844 & 8 & 43 & 9377041 & 9 & 69 & 167956256 & 5 & 54 \\
7 & 883 & 14 & 39 & 35060 & 8 & 43 & 9377041 & 9 & 69 & 167956256 & 5 & 54 \\
8 & 983 & 15 & 41 & 35060 & 8 & 43 & 9377041 & 10 & 68 & 167956256 & 5 & 54 \\
9 & 1188 & 17 & 44 & 38048 & 8 & 44 & 9377041 & 10 & 68 & 167956256 & 6 & 53 \\
10 & 1412 & 19 & 47 & 43880 & 9 & 45 & 9377041 & 10 & 68 & 167956256 & 5 & 54 \\
\hline\hline
$r$ & $N(2,6,r)$ & $x$ & $b$ & $N(3,6,r)$ & $x$ & $b$ & $N(4,6,r)$ & $x$ & $b$ & $N(5,6,r)$ & $x$ & $b$ \\
\hline
1 & 637 & 10 & 37 & 40416 & 8 & 45 & 11728881 & 10 & 72 & 191116579 & 6 & 54 \\
2 & 637 & 10 & 37 & 40416 & 8 & 45 & 11728881 & 10 & 72 & 191116579 & 6 & 54 \\
3 & 637 & 10 & 37 & 40416 & 8 & 45 & 11728881 & 10 & 72 & 191116579 & 6 & 54 \\
4 & 637 & 11 & 35 & 40416 & 8 & 45 & 11728881 & 10 & 72 & 191116579 & 6 & 54 \\
5 & 834 & 13 & 40 & 40416 & 8 & 45 & 11728881 & 10 & 72 & 191116579 & 6 & 54 \\
6 & 870 & 13 & 40 & 40416 & 8 & 45 & 11728881 & 10 & 72 & 191116579 & 6 & 54 \\
7 & 958 & 15 & 40 & 40416 & 8 & 45 & 11728881 & 10 & 72 & 191116579 & 6 & 54 \\
8 & 1163 & 17 & 43 & 41450 & 9 & 44 & 11728881 & 10 & 72 & 191116579 & 6 & 54 \\
9 & 1387 & 19 & 46 & 48066 & 9 & 47 & 11728881 & 10 & 72 & 191116579 & 6 & 54 \\
10 & 1668 & 21 & 51 & 49893 & 10 & 46 & 11728881 & 10 & 72 & 191116579 & 6 & 54 \\
\hline
\end{tabular}
}
\end{center}
\caption{Values of $N(k,n,r,\mathbb{N}$) for $n>1$ 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
$(s,t)$ & $k=2$  & $x$ & $b$ & $k=3$ & $x$ & $b$ \\ \hline
$(2,-1)$&	1923 & 18&64 & 			212595 & 15&77 \\
$(2,+1)$&2355 &20 &71 & 			266459 & 16&83  \\ \hline
$(3,-2)$&	3250 & 23& 83& 			942316 & 25&126 \\
$(3,-1)$&	3014 &22 &80 & 			957226 & 25&126 \\
$(3,+1)$&4093 & 26&92 & 			1103569 & 26&132\\
$(3,+2)$&4414 & 27&96 & 			1181758 & 27&135\\ \hline
$(4,-3)$&	10588 & 42&148 & 			2576040 & 35&174\\
$(4,-1)$&	11268 & 43&153 & 			3026615 & 37&184\\
$(4,+1)$&13708 & 48&167 & 			3152462 & 37&187 \\
$(4,+3)$&14948 & 50&175 & 			3534459 & 39&193\\ \hline
$(5,-4)$&	14900 & 50&174 & 			6146241 & 47&232\\
$(5,-3)$&	14121 & 49&170 & 			6373428 & 47&236\\
$(5,-2)$&	16810 & 53&186 & 			6672804 & 48&239\\
$(5,-1)$&	16379 & 52&184 & 			7077048 & 49&244\\
$(5,+1)$&17242 & 54&187 & 			7165274 & 49&245\\
$(5,+2)$&19090 & 57&198 & 			7526193 & 50&249\\
$(5,+3)$&19690 & 58&201 & 			7821959 & 51&252\\
$(5,+4)$&19799 & 58&201 & 			8326652 & 52&257\\ \hline
$(6,-5)$&	255964 & 209&717 & 		32025571 & 82&402\\
$(6,-1)$&	261868 & 211&727 & 		35431051 & 85&416\\
$(6,+1)$&270796 & 215&738 & 		38008681 & 87&426 \\
$(6,+5)$&282028 & 219&754 & 		40622251 & 88&436 \\ \hline
$(7,-6)$&	44329 & 87&300 & 			24233667 & 74&367 \\
$(7,-5)$&	45769 & 88&305 & 			23668124 & 74&363 \\
$(7,-4)$&	49737 & 92&317 & 			25473560 & 76&373 \\
$(7,-3)$&	49009 & 91&315 & 			26139255 & 76&376 \\
$(7,-2)$&	48989 & 91&315 & 			27035708 & 77&380 \\
$(7,-1)$&	49537 & 92&317 & 			27348027 & 77&382 \\
$(7,+1)$&51889 & 94&324 & 			28963994 & 79&389 \\
$(7,+2)$&55884 & 97&337 & 			28297320 & 78&387 \\
$(7,+3)$&54217 & 96&331 & 			30183369 & 80&394  \\
$(7,+4)$&60377 & 101&350 & 		28992218 & 79&389 \\ 
$(7,+5)$&58292 & 99&344 & 			31374203 & 81&400  \\
$(7,+6)$&63453 & 104&358 & 		31015095 & 81&397 \\ \hline
$(8,-7)$&	183828 & 177&608 & 		43603746 & 91&445  \\
$(8,-5)$&	186684 & 178&614 & 		44323025 & 91&448 \\
$(8,-3)$&	192748 & 181&623 & 		44594177 & 91&449  \\
$(8,-1)$&	199124 & 184&634 & 		49916598 & 95&466  \\
$(8,+1)$&208164 & 188&648 & 		51794250 & 96&472  \\
$(8,+3)$&216940 & 192&661 & 		53940372 & 97&479  \\
$(8,+5)$&223884 & 195&672 & 		53774817 & 97&478 \\
$(8,+7)$&227204 & 197&676 & 		55157135 & 98&482 \\ \hline
$(9,-8)$&	104873 & 134&460 & 		316621582 & 176&861  \\
$(9,-7)$&	114857 & 140&481 & 		317215246 & 176&862  \\
$(9,-5)$&	114653 & 140&481 & 		327375655 & 178&871  \\
$(9,-4)$&	118829 & 142&490 & 		329700964 & 179&872  \\
$(9,-2)$&	120113 & 143&492 & 		338139583 & 180&880  \\
$(9,-1)$&	130217 & 149&512 & 		339498184 & 180&882  \\
$(9,+1)$&134681 & 151&522 & 		352115215 & 183&891 \\
$(9,+2)$&129149 & 148&511 & 		358747834 & 184&897 \\
$(9,+4)$&137873 & 153&528 & 		371854375 & 186&908 \\
$(9,+5)$&141329 & 155&534 & 		365220868 & 185&902 \\
$(9,+7)$&142825 & 156&536 & 		383482411 & 188&917 \\
$(9,+8)$&149990 & 160&549 & 		376489804 & 187&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 $\gcd (s,t) > 1$ 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
$(s,t)$ & $k=2$  & $x$ & $b$ & $k=3$ & $x$ & $b$ \\ \hline
$(10,-1)$&2866844&701&2396&  167900541&142&698 \\
$(10,+1)$&2770803&689&2356&  164930981&142&693 \\ 
$(11,-1)$&251377&207&711&  188148921&148&724 \\
$(11,+1)$&260001&211&723&  200560127&151&740 \\
$(12,-1)$&1186948&451&1543&1871937463&320&1555 \\
$(12,+1)$&1207948&455&1556&1897625923&321&1562 \\ 
 $(13,-1)$&484333&288&986&427144568&195&951 \\
$(13,+1)$&498269&292&1000&434996727&196&957 \\
$(14,-1)$&14209388&1561&5333&718660158&232&1130 \\
$(14,+1)$&14254244&1563&5342&750996509&235&1148 \\ 
 $(15,-1)$&878885&388&1328&7192487965&501&2434 \\
$(15,+1)$&890945&390&1338&7247153841&502&2440 \\
 $(16,-1)$&4345668&863&2950&1162662009&272&1328 \\
 $(16,+1)$&4411364&869&2973&1188105593&274&1337 \\
 $(17,-1)$&1468737&501&1717&1528625985&298&1454 \\
$(17,+1)$&1487777&505&1727&1574453445&302&1468 \\
$(18,-1)$&47752420&2862&9774&23390399911&742&3606 \\
$(18,+1)$&47891524&2866&9789&23431535880&743&3607 \\ 
$(19,-1)$&2296953&627&2146&2670453204&360&1750 \\
$(19,+1)$&2330393&632&2161&2654207231&359&1746 \\
$(20,-1)$&12065164&1438&4915&3392160594&390&1895 \\
$(20,+1)$&12241324&1449&4950&3426870488&391&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 $\gcd (s,t) > 1$ 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 $k>1$.

\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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E. Zeckendorf, Repr\'{e}sentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, \textit{Bull. Soc. Roy. Sci. Li\`{e}ge} \textbf{41} (1972), 179--182.

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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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\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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