1
2
3
47
6
Journal of Integer Sequences, Vol. 18 (2015),
Article 15.10.5
23 11
Generalized Anti-Waring Numbers
Chris Fuller and Robert H. Nichols, Jr.
Labry School of Science, Technology, and Business
Cumberland University
1 Cumberland Square
Lebanon, TN 37087
USA
cfuller@cumberland.edu
rnichols@cumberland.edu
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 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.
1
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 th powers of r or more distinct positive integers. Several authors [3, 5, 7, 11]
recently reported results on anti-Waring numbers.
Early results considered only r = 1. As early as 1948, Sprague found that N (2, 1) =
129 [15] and proved that N (k, 1) exists for all k ≥ 2 [16]. In 1964, Graham [6] reported
that N (3, 1) = 12759 (Graham [6] references another Graham paper “On the Threshold of
completeness for certain sequences of polynomial values” said to appear circa 1964). Dressler
and Parker [4] also computed N (3, 1) in 1974. Lin [10] used Graham’s method to find that
1
N (4, 1) = 5134241 with a computer in 1970. In 1992, Patterson [12, pp. 18–23] 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 [11] proved that N (k, r) exists for all positive integers
k and r. Deering and Jamieson [3] found specific values of N (2, r) for 1 ≤ r ≤ 10 and N (3, r)
for 1 ≤ r ≤ 5. Shortly afterwards, Fuller et al. [5] computed values of N (2, r) for 1 ≤ r ≤ 50
and N (3, r) for 1 ≤ r ≤ 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
[3, 5].
In an effort to generalize the anti-Waring results we consider a nondecreasing sequence
of positive integers A = (ai )i∈N . Here and throughout we use N = {1, 2, 3, . . . }. For positive
integers k, n, and r we define the 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 th powers of the ai with i ≥ 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 1 and 2 in Section 2). Looper
and Saritzky [11] proved that both the anti-Waring number N (k, r) and the generalized
anti-Waring number N (k, n, r, 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 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 threshold of completeness, θ(S), is the largest positive integer that is
not expressible as a sum of distinct elements of S. Therefore, the threshold of completeness,
θ(S), is one less than the generalized anti-Waring number N (1, 1, 1, S). Also, if S = (si )i∈N is
a nondecreasing sequence of positive integers such that the sequence (ski )i≥n is complete, then
the generalized anti-Waring number N (k, n, 1, S) exists and N (k, n, 1, S) − 1 = θ (ski )i≥n .
Brown [1] 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 [9] reported an account of the Zeckendorf representation (circa 1939 [17]), 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,
Kløve [8] found thresholds of completeness for sequences of the form (⌊iα ⌋)i∈N , where ⌊x⌋ is
the floor function, for 1 ≤ α ≤ 4.18 in increments of 0.02. In 1978, Porubský [13] proved that
N (k, 1, 1, P) exists for all positive integers k and the sequence of primes P. Burr and Erdős
[2] 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 N. The generalization also extends the concept of complete sequences
2
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.
2
Verifying N (k, n, r, A), when it exists
For given positive integers k, n, r, and any nondecreasing sequence of positive integers
A = (ai )i∈N , we define a positive integer to be (k, n, r, A)-good if it can be written as a sum
of the k th powers of r or more distinct elements of the sequence (ai )i≥n . We define a positive
integer that is not (k, n, r, A)-good to be (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, . . .) of positive even integers will
never be odd. This is an instance of a more general phenomenon.
Theorem 1. Let A = (ai )i∈N be a nondecreasing sequence of positive integers. If all ai for
i ≥ 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.
Proof. Every sum of positive powers of the ai , i ≥ 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 an , an+1 , . . . also exist.
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.
Theorem 2. Let A = (i!)i∈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.
Proof. First notice that for each ai ∈ A,


if i = 1;
1,
k
k
ai mod 6 ≡ 2 mod 6, if i = 2;


0,
if i > 2.
Consider any (k, n, r, A)-good number m. Distinct integers i1 , i2 , . . . , it exist such that
m = aki1 + aki2 + · · · + akit
where t ≥ r and iα ≥ n for each α ∈ {1, 2, . . . , t}. Thus the sum m must be 0, 1, 2k , or
1 + 2k 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.
3
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, N) exist for all k, n, and
r [11]. 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 [1,
Theorem 1] in terms of generalized anti-Waring numbers.
Theorem 3. Let k and n be positive integers, and let A = (ai )i∈N be a nondecreasing sequence
of positive integers. The generalized anti-Waring number N (k, n, 1, A) both
and equals
Pp exists
k
k
one if and only if (i) an = 1 and (ii) for all integers p ≥ n, ap+1 ≤ 1 + i=n ai .
This result only considers r = 1. Also since Brown [1] defined complete sequences
requiring the threshold of completeness to be zero, he requires an = 1. Theorem 3 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 20 ). We
must consider different conditions for the more general definition of complete sequences with
any threshold of completeness.
The next result from Graham [6, Theorem 4] establishes completeness conditions for
sequences generated by polynomials.
Theorem 4. Let f (x) be a polynomial with real coefficients expressed in the form
x
x
f (x) = α0 + α1
+ · · · + αn
, αn 6= 0.
1
n
The sequence S(f ) = (f (1), f (2), · · · ) is complete if and only if
1. αk = pk /qk for some integers pk and qk with gcd(pk , qk ) = 1 and qk 6= 0 for 0 ≤ k ≤ n,
2. αn > 0, and
3. gcd(p0 , p1 , . . . , pn ) = 1.
Again, in terms of generalized anti-Waring numbers Theorem 4 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), . . .) is
complete if and only if (f (n), f (n + 1), f (n + 2), . . .) is complete for any n. The next theorem
shows that nothing like this can be expected in general.
Theorem 5. 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 ∈ {1, 2, . . . , n − 1} and N (k, j, r, A) ≤ N (k, n, r, A). Furthermore, the
converse is false.
4
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 th powers of r or more distinct elements of (ai )i≥n , then, with the
same elements, each positive integer can be written as a sum k th powers of r or more distinct
elements of (ai )i≥j for j ∈ {1, 2, . . . , n − 1}. Therefore, we have N (k, j, r, A) ≤ N (k, n, r, A)
for j ∈ {1, 2, . . . , n − 1}.
To see that the converse is false, consider the sequence A = (2i−1 )i∈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 (2i−1 )i≥2 .
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 6 is a generalization of a recent result
for anti-Waring numbers [5, Theorem 2.2].
Theorem 6. Let k, n, r, b, and N̂ be positive integers, and let A = (ai )i∈N with 0 < ai ≤ ai+1
and ai ∈ N for all i. If the consecutive integers {N̂ , . . . , bk } are all (k, n, r, A)-good, the
number N̂ − 1 is (k, n, r, A)-bad, and there exists a positive integer x such that the conditions
1. N̂ ≤ bk + 1 − (b − x)k ,
2. an ≤ b − x,
3. 0 <
n+r−2
X
i=n
aki
!
+ 2(m − x)k − (m + 1)k for all m ≥ b, and
4. (m + 1)k − (m − x)k ≤ mk for all m ≥ b
hold, then the generalized anti-Waring number N (k, n, r, A) exists and equals N̂ . Note: The
sum in condition 3 is zero if r = 1.
Proof. We want to prove that if ℓ ≤ mk and ℓ is (k, n, r, A)-bad, then ℓ ≤ N̂ − 1 by induction
on m with m ≥ b.
This is clearly true for m = b as we know the consecutive integers {N̂ , . . . , bk } are all
(k, n, r, A)-good.
Now suppose ℓ ≤ (m+1)k and ℓ is (k, n, r, A)-bad. If ℓ ≤ mk , then by induction ℓ ≤ N̂ −1.
Next, consider ℓ such that
mk + 1 ≤ ℓ ≤ (m + 1)k .
(1)
Notice bk − (b − x)k ≤ mk − (m − x)k for m ≥ b. Using this along with (1) and condition 1,
we have
N̂ ≤ ℓ − (m − x)k .
(2)
To see that ℓ − (m − x)k is (k, n, r, A)-bad, suppose it is (k, n, r, A)-good. Then
ℓ − (m − x)k = aki1 + aki2 + aki3 + · · · + akit
5
where t ≥ r, iα 6= iβ for all α 6= β, and iα ≥ n for all α ∈ {1, 2, . . . , t}. Since ℓ is (k, n, r, A)bad and
ℓ = aki1 + aki2 + aki3 + · · · + akit + (m − x)k ,
either m − x < an , which contradicts condition 2, or aiα = m − x for some α ∈ {1, 2, . . . , t}.
Therefore,
ℓ ≥ akn + akn+1 + akn+2 + · · · + akn+r−2 + 2(m − x)k .
If r = 1, this is just ℓ ≥ 2(m − x)k . Combining with (1), we get
!
n+r−2
X
aki + 2(m − x)k − (m + 1)k ≤ 0.
i=n
This contradicts condition 3 and means that ℓ − (m − x)k must be (k, n, r, A)-bad.
Now from (1) and condition 4,
ℓ − (m − x)k ≤ (m + 1)k − (m − x)k ≤ mk .
By induction we then have ℓ − (m − x)k ≤ N̂ − 1. This contradicts (2). Hence there are no
ℓ that are (k, n, r, A)-bad and satisfy (1).
Most of the threshold of completeness results in the literature of complete sequences
rely on work by Richert [14], 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 6.
3
Values of N (k, n, r, A)
As a result of Theorems 1 and 2, 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 6.
Johnson and Laughlin [7, Theorem 1] proved a first result
N (1, 1, r, N) =
r
X
i=1
r
i = (r + 1)
2
(3)
for the case of k = n = 1. A similar argument is valid for general values of n.
Theorem 7. For positive integers n and r, the generalized anti-Waring number is given by
N (1, n, r, N) =
n+r−1
X
i=n
r
i = (r + 1) + r(n − 1).
2
6
P
Proof. Clearly, the sum n+r−1
i is the smallest integer expressible as the sum of r or more
i=n
distinct
integers
greater
than
or
equal to n. For any positive integer x greater than the sum
Pn+r−1
i, we have
i=n
n+r−2
X
x−
i > n + r − 1.
i=n
Finally, we have that
x=
n+r−2
X
i+
x−
i=n
n+r−2
X
i=n
i
!
so the integer x is the sum of r distinct integers greater than or equal to n.
Theorem 8. 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
N (1, n, r, (si + t)i∈N ) = 1 − s +
n+r+s−2
X
(si + t).
(4)
i=n
Note: For the case of s = 1 and t = 0, this reduces to N (1, n, r, N) and agrees with
Theorem 7.
Proof. The sequence B = (si + t)i≥n consists of all positive integers equivalent to t mod 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 mod 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 Zs . The list r, r + 1, r + 2, . . . , r + s − 1 contains representatives of
each equivalence class in Zs . Since the integers s and t are relatively prime, the same is true
for the list rt, (r + 1)t, (r + 2)t, . . . , (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 ∈ {r, r + 1, r + 2, . . . , r + s − 1}, let mp be the sum of the first p elements of B, i.e.,
!
n+p−1
n+p−1
X
X
i + pt.
(si + t) = s
mp =
i=n
i=n
As noted before, we have mp ≡ pt (mod s). We also know that mp is the smallest integer
equivalent to pt mod s expressible as the sum of r or more distinct elements of B. Hence
the integer mp − s is (1, n, r, (si + t)i∈N )-bad. If a positive integer x ≥ mp is also equivalent
to pt mod s, then we have x = mp + ℓs for some positive ℓ ∈ Z or, equivalently,
x = ℓs +
n+p−1
n+p−2
X
X
(si + t) = (s(ℓ + n + p − 1) + t) +
i=n
i=n
7
(si + t).
Thus, all integers equivalent to pt mod s greater than mp are expressible as the sum of r or
more distinct elements of B. Since we have mp < mp+1 for all p, the last (1, n, r, (si + t)i∈N )bad integer is mr+s−1 − s. Therefore, the generalized anti-Waring number is N (1, n, r, (si +
t)i∈N ) = mr+s−1 − s + 1 which is (4).
k
1
2
3
4
5
6
N (k, 1)
1
129
12759
5134241
67898772
11146309948
x
1
4
5
8
4
5
b
4
18
32
59
45
55
bad count
0
31
2788
889576
13912682
2037573096
Table 1: Values of N (k, 1, 1, N)
k
1
2
3
N (k, 1, 1, P)
7
17164
1866001
x
6
54
31
b
14
187
157
bad count
3
2438
483370
Table 2: Values of N (k, 1, 1, P)
For most cases, a formula for N (k, n, r, A) is not known, but we can compute particular
values. In the Tables 1 to 6 we list values of N (k, n, r, A) along with the corresponding x
and b that satisfy the conditions for Theorem 6 hence confirming the given generalized antiWaring number. Tables 1, 3, and 4 use A = N. In Table 1 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 th
powers of distinct integers. For each k we also include a bad count, i.e., the number of positive
integers that cannot be written as a sum of k th powers. Table 2 lists the corresponding values
over the sequence of primes P. Table 3 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, N) for much larger r. For example, we found that N (2, 1, 1000, N) = 333951595
with x = 12898 and b = 19395. Table 4 lists generalized anti-Waring numbers for varying
k, n, and r. Tables 3 and 4 omit generalized anti-Waring numbers when k = 1 because
a formula for N (1, n, r, N) for all n and r in N exists by Theorem 7. Tables 5 and 6 list
generalized anti-Waring numbers for fixed n = 1 and r = 1 over various sequences of the
form (si + t)i∈N .
8
r
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
N (2, r)
129
129
129
129
198
238
331
383
528
648
889
989
1178
1398
1723
1991
2312
2673
3048
3493
4094
4614
5139
5719
6380
7124
7953
8677
9538
10394
11559
12603
13744
14864
16253
17529
x
4
4
4
4
6
6
8
9
10
12
14
15
17
19
21
24
26
28
31
34
36
39
42
44
48
51
54
57
61
63
67
71
74
78
81
85
b
18
18
18
18
22
23
26
27
32
33
39
41
44
47
52
54
58
62
65
69
75
79
83
87
91
96
101
105
109
114
120
125
130
135
141
146
N (3, r)
12759
12759
12759
12759
12759
15279
15279
15279
16224
18149
22398
24855
28887
36951
39660
49083
56076
66534
75912
87567
101093
122064
138696
156498
179520
201921
227400
256254
289869
325590
359358
401496
448503
496257
554217
611736
x
5
5
5
5
5
6
6
6
6
6
7
7
8
9
9
10
11
12
12
13
14
15
16
17
18
19
20
22
23
24
25
26
27
29
30
30
N (4, r)
5134241
5134241
5134241
5134241
5134241
5134241
5134241
5134241
5134241
5134241
5191473
5626194
6018930
6408466
6664722
6938867
8077523
8592323
9269124
10418260
10589380
12852837
13199973
15148358
16526214
17803895
20499591
21202776
24306872
25670088
29819129
31126025
35677050
38187306
43256507
46180043
b
32
32
32
32
32
33
33
33
33
35
37
38
39
42
43
46
47
50
52
54
56
60
62
64
67
69
72
73
76
79
82
85
88
90
93
97
x
8
8
8
8
8
8
8
8
8
8
8
8
8
9
9
9
9
9
10
10
10
11
11
11
12
12
13
13
13
14
14
15
15
16
16
17
b
59
59
59
59
59
59
59
59
59
59
59
60
62
62
62
63
66
67
67
69
70
72
73
76
76
78
81
81
84
84
88
88
92
92
96
97
N (5, r)
67898772
67898772
67898772
67898772
67898772
67898772
67898772
67898772
67898772
67898772
67898772
67898772
71780055
74729904
81846431
92894512
95723448
112031630
124811198
142118181
163637305
189572962
210715205
247073537
285744830
319712379
374237223
430026890
491665093
558015873
640101337
737104155
839165455
950792455
1070200765
1215652918
x
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
9
9
9
10
10
b
45
45
45
45
45
45
45
45
45
45
45
45
46
46
45
47
47
49
50
52
52
54
55
57
57
59
61
63
64
65
68
68
71
73
73
76
Table 3: Values of N (k, 1, r, N) and the corresponding x and b that satisfy Theorem 6. Values
of N (1, n, r, N) are given by Theorem 7.
9
r
1
2
3
4
5
6
7
8
9
10
r
1
2
3
4
5
6
7
8
9
10
r
1
2
3
4
5
6
7
8
9
10
r
1
2
3
4
5
6
7
8
9
10
r
1
2
3
4
5
6
7
8
9
10
N (2, 2, r)
193
193
193
213
318
334
398
527
647
888
N (2, 3, r)
224
224
233
314
330
418
523
643
884
984
N (2, 4, r)
385
385
385
385
453
558
634
875
999
1164
N (2, 5, r)
493
493
493
494
542
670
883
983
1188
1412
N (2, 6, r)
637
637
637
637
834
870
958
1163
1387
1668
x
5
5
5
6
7
8
9
10
12
14
x
6
6
6
7
8
9
10
12
14
15
x
8
8
8
8
9
10
12
14
15
17
x
9
9
9
9
10
12
14
15
17
19
x
10
10
10
11
13
13
15
17
19
21
b
22
22
22
22
27
26
27
32
33
39
b
23
23
23
26
26
28
32
33
39
41
b
30
30
29
28
30
33
34
39
41
43
b
34
33
32
32
33
35
39
41
44
47
b
37
37
37
35
40
40
40
43
46
51
N (3, 2, r)
19310
19310
19310
19310
19310
19310
19310
19310
20885
24098
N (3, 3, r)
23775
23775
23775
23775
23775
23775
23775
24756
28221
28950
N (3, 4, r)
26862
26862
26862
26862
26862
26862
27528
28194
30111
33234
N (3, 5, r)
34844
34844
34844
34844
34844
34844
35060
35060
38048
43880
N (3, 6, r)
40416
40416
40416
40416
40416
40416
40416
41450
48066
49893
x
6
6
6
6
6
6
6
6
7
7
x
7
7
7
7
7
7
7
7
7
8
x
7
7
7
7
7
7
7
7
8
8
x
8
8
8
8
8
8
8
8
8
9
x
8
8
8
8
8
8
8
9
9
10
b
36
36
36
36
36
36
36
36
36
38
b
38
38
38
38
38
38
38
38
41
40
b
40
40
40
40
40
40
40
41
40
42
b
43
43
43
43
43
43
43
43
44
45
b
45
45
45
45
45
45
45
44
47
46
N (4, 2, r)
6659841
6659841
6659841
6659841
6659841
6692881
6692881
6692881
6778897
6778897
N (4, 3, r)
7076321
7076321
7076321
7076321
7076321
7076321
7103505
7103505
7103505
7103505
N (4, 4, r)
8912545
8912545
8912545
8912545
8912545
8912545
8912545
8912545
8912545
8912545
N (4, 5, r)
9292705
9292705
9292705
9292705
9292705
9377041
9377041
9377041
9377041
9377041
N (4, 6, r)
11728881
11728881
11728881
11728881
11728881
11728881
11728881
11728881
11728881
11728881
x
9
9
9
9
9
9
9
9
9
9
x
9
9
9
9
9
9
9
9
9
9
x
9
9
9
9
9
9
9
9
9
9
x
9
9
9
9
9
9
9
10
10
10
x
10
10
10
10
10
10
10
10
10
10
b
62
62
62
62
62
62
62
62
62
62
b
63
63
63
63
63
63
63
63
63
63
b
68
68
68
68
68
68
68
68
68
68
b
69
69
69
69
69
69
69
68
68
68
b
72
72
72
72
72
72
72
72
72
72
N (5, 2, r)
84038312
84038312
84038312
84038312
84038312
84038312
84038312
84038312
84038312
84038312
N (5, 3, r)
110100822
110100822
110100822
110100822
110100822
110100822
110100822
110100822
110100822
110100822
N (5, 4, r)
129436797
129436797
129436797
129436797
129436797
129436797
129436797
129436797
129436797
130964972
N (5, 5, r)
167956256
167956256
167956256
167956256
167956256
167956256
167956256
167956256
167956256
167956256
N (5, 6, r)
191116579
191116579
191116579
191116579
191116579
191116579
191116579
191116579
191116579
191116579
x
5
5
5
5
5
5
5
5
5
5
x
5
5
5
5
5
5
5
5
5
5
x
5
5
5
5
5
5
5
5
5
5
x
5
5
5
5
5
5
5
5
6
5
x
6
6
6
6
6
6
6
6
6
6
b
46
46
46
46
46
46
46
46
46
46
b
49
49
49
49
49
49
49
49
49
49
b
51
51
51
51
51
51
51
51
51
51
b
54
54
54
54
54
54
54
54
53
54
b
54
54
54
54
54
54
54
54
54
54
Table 4: Values of N (k, n, r, N) for n > 1 and the corresponding x and b that satisfy
Theorem 6. Values of N (1, n, r, N) are given by Theorem 7.
10
(s, t)
(2, −1)
(2, +1)
(3, −2)
(3, −1)
(3, +1)
(3, +2)
(4, −3)
(4, −1)
(4, +1)
(4, +3)
(5, −4)
(5, −3)
(5, −2)
(5, −1)
(5, +1)
(5, +2)
(5, +3)
(5, +4)
(6, −5)
(6, −1)
(6, +1)
(6, +5)
(7, −6)
(7, −5)
(7, −4)
(7, −3)
(7, −2)
(7, −1)
(7, +1)
(7, +2)
(7, +3)
(7, +4)
(7, +5)
(7, +6)
(8, −7)
(8, −5)
(8, −3)
(8, −1)
(8, +1)
(8, +3)
(8, +5)
(8, +7)
(9, −8)
(9, −7)
(9, −5)
(9, −4)
(9, −2)
(9, −1)
(9, +1)
(9, +2)
(9, +4)
(9, +5)
(9, +7)
(9, +8)
k=2
1923
2355
3250
3014
4093
4414
10588
11268
13708
14948
14900
14121
16810
16379
17242
19090
19690
19799
255964
261868
270796
282028
44329
45769
49737
49009
48989
49537
51889
55884
54217
60377
58292
63453
183828
186684
192748
199124
208164
216940
223884
227204
104873
114857
114653
118829
120113
130217
134681
129149
137873
141329
142825
149990
x
18
20
23
22
26
27
42
43
48
50
50
49
53
52
54
57
58
58
209
211
215
219
87
88
92
91
91
92
94
97
96
101
99
104
177
178
181
184
188
192
195
197
134
140
140
142
143
149
151
148
153
155
156
160
b
64
71
83
80
92
96
148
153
167
175
174
170
186
184
187
198
201
201
717
727
738
754
300
305
317
315
315
317
324
337
331
350
344
358
608
614
623
634
648
661
672
676
460
481
481
490
492
512
522
511
528
534
536
549
k=3
212595
266459
942316
957226
1103569
1181758
2576040
3026615
3152462
3534459
6146241
6373428
6672804
7077048
7165274
7526193
7821959
8326652
32025571
35431051
38008681
40622251
24233667
23668124
25473560
26139255
27035708
27348027
28963994
28297320
30183369
28992218
31374203
31015095
43603746
44323025
44594177
49916598
51794250
53940372
53774817
55157135
316621582
317215246
327375655
329700964
338139583
339498184
352115215
358747834
371854375
365220868
383482411
376489804
x
15
16
25
25
26
27
35
37
37
39
47
47
48
49
49
50
51
52
82
85
87
88
74
74
76
76
77
77
79
78
80
79
81
81
91
91
91
95
96
97
97
98
176
176
178
179
180
180
183
184
186
185
188
187
b
77
83
126
126
132
135
174
184
187
193
232
236
239
244
245
249
252
257
402
416
426
436
367
363
373
376
380
382
389
387
394
389
400
397
445
448
449
466
472
479
478
482
861
862
871
872
880
882
891
897
908
902
917
911
Table 5: Values of N (k, 1, 1, (si+t)i∈N ) and the corresponding x and b that satisfy Theorem 6.
The generalized anti-Waring number N (k, n, r, (si + t)i∈N ) does not exist if gcd(s, t) > 1 by
Theorem 1, and values of N (1, n, r, (si + t)i∈N ) are given by Theorem 8.
11
(s, t)
(10, −1)
(10, +1)
(11, −1)
(11, +1)
(12, −1)
(12, +1)
(13, −1)
(13, +1)
(14, −1)
(14, +1)
(15, −1)
(15, +1)
(16, −1)
(16, +1)
(17, −1)
(17, +1)
(18, −1)
(18, +1)
(19, −1)
(19, +1)
(20, −1)
(20, +1)
k=2
2866844
2770803
251377
260001
1186948
1207948
484333
498269
14209388
14254244
878885
890945
4345668
4411364
1468737
1487777
47752420
47891524
2296953
2330393
12065164
12241324
x
701
689
207
211
451
455
288
292
1561
1563
388
390
863
869
501
505
2862
2866
627
632
1438
1449
b
2396
2356
711
723
1543
1556
986
1000
5333
5342
1328
1338
2950
2973
1717
1727
9774
9789
2146
2161
4915
4950
k=3
167900541
164930981
188148921
200560127
1871937463
1897625923
427144568
434996727
718660158
750996509
7192487965
7247153841
1162662009
1188105593
1528625985
1574453445
23390399911
23431535880
2670453204
2654207231
3392160594
3426870488
x
142
142
148
151
320
321
195
196
232
235
501
502
272
274
298
302
742
743
360
359
390
391
b
698
693
724
740
1555
1562
951
957
1130
1148
2434
2440
1328
1337
1454
1468
3606
3607
1750
1746
1895
1901
Table 6: Additional values of N (k, 1, 1, (si + t)i∈N ) and the corresponding x and b that
satisfy Theorem 6. The generalized anti-Waring number N (k, n, r, (si + t)i∈N ) does not exist
if gcd(s, t) > 1 by Theorem 1, and values of N (1, n, r, (si + t)i∈N ) are given by Theorem 8.
4
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 ≤ j we
have the inequalities N (k, i, r, A) ≤ N (k, j, r, A) and N (k, n, i, A) ≤ N (k, n, j, A) when each
exists. We are unable to prove the inequality N (k, n, r, A) ≤ 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.
5
Acknowledgments
We thank the editor and the anonymous referee for their time and consideration. The
referee’s report was thorough and included valuable suggestions.
12
References
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(1961), 557–560.
[2] S. Burr and P. Erdős, Completeness properties of perturbed sequences, J. Number Theory 13 (1981), 446–455.
[3] J. Deering and W. Jamieson, On anti-Waring numbers, to appear in J. Combin. Math.
Combin. Comput.
[4] R. Dressler and T. Parker, 12,758, Math. Comp. 28 (1974), 313–314.
[5] C. Fuller, D. Prier, and K. Vasconi, New results on an anti-Waring problem, Involve 7
(2014), 239–244.
[6] R. L. Graham, Complete sequences of polynomial values, Duke Math. J. 31 (1964),
275–285.
[7] P. Johnson and M. Laughlin, An anti-Waring conjecture and problem, Int. J. Math.
Comput. Sci. 6 (2011), 21–26.
[8] T. Kløve, Sums of distinct elements from a fixed set, Math. Comp. 29 (1975), 1144–1149.
[9] C. G. Lekkerkerker, Voorstelling van natuurlikjke getallen door een som van getallen
van Fibonaaci, Simon Stevin 29 (1952), 190–195.
[10] S. Lin, Computer experiments on sequences which form integral bases, in J. Leech, ed.,
Computational Problems in Abstract Algebra, Pergamon Press, 1970, pp. 365–370.
[11] N. Looper and N. Saritzky, An anti-Waring theorem and proof, to appear in J. Combin.
Math. Combin. Comput.
[12] C. Patterson, The Derivation of a High Speed Sieve Device, Ph.D. thesis, University of
Calgary, 1992.
[13] Š. Porubský, Sums of prime powers, Monatsh. Math 86 (1979), 301–303.
[14] H. E. Richert, Über Zerlegungen in paarweise verschiedene Zahlen, Nordisk Mat. Tidskr.
31 (1949), 120–122.
[15] R. Sprague, Über Zerlegungen in ungleiche Quadratzahlen, Math. Z. 51 (1948), 289–290.
[16] R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen,
Math. Z. 51 (1948), 466–468.
13
[17] E. Zeckendorf, Représentation des nombres naturels par une somme de nombres de
Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41 (1972), 179–182.
2010 Mathematics Subject Classification: Primary 11P05; Secondary 05A17.
Keywords: complete sequence, sum of powers, anti-Waring number.
(Concerned with sequence A001661.)
Received June 18 2015; revised versions received September 13 2015; September 21 2015.
Published in Journal of Integer Sequences, September 24 2015.
Return to Journal of Integer Sequences home page.
14