A030052 -id:A030052 - cp's OEIS frontend

A001661 Largest number not the sum of distinct positive n-th powers.

128, 12758, 5134240, 67898771, 11146309947, 766834015734, 4968618780985762

Offset: 2

a(8) > 74^8. - Donovan Johnson, Nov 23 2010
Fuller and Nichols prove that a(6) = 11146309947 and that 2037573096 positive numbers cannot be written as the sum of distinct 6th powers. - Robert Nichols, Sep 09 2017
a(8) >= 83^8 ~ 2.25e15 since A030052(8) = 84. Similarly, a(9..15) >= (46^9, 62^10, 67^11, 80^12, 101^13, 94^14, 103^15) ~ (9.2e14, 8.4e17, 1.2e20, 6.9e22, 1.1e26, 4.2e27, 1.6e30), cf. formula. Most often a(n) will be closer to and even larger than A030052(n)^n. - In the literature, a(n)+1 is known as the anti-Waring number N(n,1). - M. F. Hasler, May 15 2020
a(9..16) > (1.55e17, 1.31e19, 1.64e21, 5.55e23, 1.32e26, 1.37e28, 2.09e30, 9.99e35). - Michael J. Wiener, Jun 10 2023

S. Lin, Computer experiments on sequences which form integral bases, pp. 365-370 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. E. Dressler and T. Parker, 12,758, Math. Comp. 28 (1974), 313-314.
Shalosh B. Ekhad and Doron Zeilberger, Automating John P. D'Angelo's method to study Complete Polynomial Sequences, arXiv:2111.02832 [math.NT], 2021.
Mauro Fiorentini, Rappresentazione di interi come somma di potenze (in Italian).
C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18 (2015), #15.10.5.
R. L. Graham, Complete sequences of polynomial values, Duke Math. J. 31 (1964), pp. 275-285.
D. Kim, On the largest integer that is not a sum of distinct nth powers of positive integers, arXiv:1610.02439 [math.NT], 2016-2017.
D. Kim, On the largest integer that is not a sum of distinct nth powers of positive integers, J. Int. Seq. 20 (2017), #17.7.5.
P. LeVan and D. Prier, Improved Bounds on the Anti-Waring Number, J. Int. Seq. 20 (2017), #17.8.7.
D. C. Mayer, Sharp bounds for the partition function of integer sequences, BIT 27 (1987), 98-110.
D. C. Mayer, Partition functions via bit list operations, 2009.
N. J. A. Sloane and R. E. Dressler, Correspondence, June 1974
R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.
Eric Weisstein's World of Mathematics, Waring's Problem
M. J. Wiener, The Largest Integer Not the Sum of Distinct 8th Powers, J. Integer Sequences, 26 (2023), Article 23.5.4.
J. W. Wrench, Jr., Letter to N. J. A. Sloane, 10 Apr, 1974

Cf. A030052, A173563, A279529.

Cf. A121571 (primes instead of integers).

a(n) < d*2^(n-1)*(c*2^n + (2/3)*d*(4^n - 1) + 2*d - 2)^n + c*d, where c = n!*2^(n^2) and d = 2^(n^2 + 2*n)*c^(n-1) - 1, according to Kim [2016-2017]. - Danny Rorabaugh, Oct 11 2016

a(n) >= (A030052(n)-1)^n. - M. F. Hasler, May 15 2020

a(7) from Donovan Johnson, Nov 23 2010

a(8) from Michael J. Wiener, Jun 10 2023

A332065 Infinite square array where row n lists the integers whose n-th power is the sum of distinct n-th powers of positive integers; read by falling antidiagonals.

Original entry on oeis.org

3, 4, 5, 5, 7, 6, 6, 9, 9, 15, 7, 10, 12, 25, 12, 8, 11, 13, 27, 23, 25, 9, 12, 14, 29, 24, 28, 40, 10, 13, 15, 30, 28, 32, 43, 84, 11, 14, 16, 31, 29, 34, 44, 85, 47, 12, 15, 17, 33, 30, 35, 45, 86, 49, 63, 13, 16, 18, 35, 31, 36, 46, 87, 52, 64, 68

Offset: 1

M. F. Hasler, Mar 31 2020

Each row contains all sufficiently large integers (Sprague). Sequences A001422, A001476, A046039, A194768, A194769, ... mention the largest number which can't be written as sum of distinct n-th powers for n = 2, 3, 4, 5, 6, ...; see also A001661. Sequence A332066 gives the number of positive integers not in row n.

All positive multiples of any T(n,k) appear later in that row (because if s^n = Sum_{x in S} x^n, then (k*s)^n = Sum_{x in k*S} x^n).

			The table reads: (Entries from where on T(n,k+1) = T(n,k)+1 are marked by *.)
   n | k=1    2    3    4    5    6    7    8    9   10   11   12   13  ...
  ---+---------------------------------------------------------------------
   1 |   3*   4    5    6    7    8    9   10   11   12   13   14   15  ...
   2 |   5    7    9*  10   11   12   13   14   15   16   17   18   19  ...
   3 |   6    9   12*  13   14   15   16   17   18   19   20   21   22  ...
   4 |  15   25   27   29   30   31   33   35   37   39   41   43   45* ...
   5 |  12   23   24   28*  29   30   31   32   33   34   35   36   37  ...
   6 |  25   28   32   34*  35   36   37   38   39   40   41   42   43  ...
   7 |  40   43*  44   45   46   47   48   49   50   51   52   53   54  ...
   8 |  84*  85   86   87   88   89   90   91   92   93   94   95   96  ...
   9 |  47   49   52*  53   54   55   56   57   58   59   60   61   62  ...
  10 |  63*  64   65   66   67   68   69   70   71   72   73   74   75  ...
  11 |  68   73*  74   75   76   77   78   79   80   81   82   83   84  ...
  ...| ...
Row 1: 3^1 = 2^1 + 1^1, 4^1 = 3^1 + 1^1, 5^1 = 4^1 + 1^1, 6^1 = 5^1 + 1^1, ...
Row 2: 5^2 = 4^2 + 3^2, 7^2 = 6^2 + 3^2 + 2^2, 9^2 = 8^2 + 4^2 + 1^2, ...
Row 3: 6^3 = 5^3 + 4^3 + 3^3, 9^3 = 8^3 + 6^3 + 1, 12^3 = 10^3 + 8^3 + 6^3, ...
Row 4: 15^4 = Sum {14, 9, 8, 6, 4}^4, 25^4 = Sum {21, 20, 12, 10, 8, 6, 2}^4, ...
See the link for other rows.

R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.
Various authors, Decomposition of T(n,1)^n = A030052(n)^n.

Cf. A030052 (first column), A001661.
Cf. A009003 (hypotenuse numbers; subsequence of row 2).
Cf. A001422, A001476, A046039, A194768, A194769.
Cf. A332066.

M332065=Map(); A332065(n,m,r)={if(r, if( m<2^n||m>r^n*(r+n+1)\(n+1), m<2, r=min(sqrtnint(m,n),r), m==r^n || while( !A332065(n,m-r^n,r-=1) && (mA004736(n),n=A002260(n)]; mapisdefined(M332065,[n,m],&r), r, n<2, m+2, r=if(m>1,A332065(n,m-1),n+2); until(A332065(n, (r+=1)^n, r-1),); mapput(M332065,[n,m],r); r)} \\ Calls itself with nonzero (optional) 3rd argument to find by exhaustive search whether r can be written as sum of distinct powers <= m^n. (Comment added by M. F. Hasler, May 25 2020)

T(1,k) = 2 + k for all k. (Indeed, s^1 = (s-1)^1 + 1 and s-1 > 1 for s > 2.)
T(2,k) = 6 + k for all k >= 3. (Use s^2 = (s-1)^2 + 2*s-1 and A001422, A009003.)
T(3,k) = 9 + k for all k >= 3. (Use max A001476 = 12758 < 24^3.)
T(4,k) = 32 + k for all k >= 13. (Use max A046039 < 48^4.)
T(5,k) = 24 + k for all k >= 4. (Use max(N \ A194768) < 37^5.)
T(6,k) = 30 + k for all k >= 4. (Use max(N \ A194769) < 48^6.)
T(7,k) = 41 + k for all k >= 2.
T(9,k) = 49 + k for all k >= 3.

More terms from M. F. Hasler, Jul 19 2020

A078607 Least positive integer x such that 2*x^n > (x+1)^n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 40, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102

Offset: 0

Jon Perry, Dec 09 2002

nonn

Also, integer for which E(s) = s^n - Sum_{0 < k < s} k^n is maximal. It appears that a(n) + 2 is the least integer for which E(s) < 0. - M. F. Hasler, May 08 2020

			a(2) = 3 as 2^2 = 4, 3^2 = 9 and 4^2 = 16.
For n = 777451915729368, a(n) = 1121626023352384 = ceiling(n log 2), where n*log(2) = 1121626023352383.5 - 2.13*10^-17 and 2*floor(n log 2)^n / floor(1 + n log 2)^n = 1 - 3.2*10^-32. - _M. F. Hasler_, Nov 02 2013
a(2) is given by floor(1/(1-1/sqrt(2))). [From former A230748.]

Vincenzo Librandi, Table of n, a(n) for n = 0..20000

Cf. A224996 (the largest integer x that satisfies 2*x^n <= (x+1)^n).
Cf. A050499, A050500.
Cf. A078608, A078609. Equals A110882(n)-1 for n > 0.
Cf. A332097 (maximum of E(s), cf comments), also related to this:  A332101 (least k such that k^n <= sum of all smaller n-th powers), A030052 (least k such that k^n = sum of distinct n-th powers), A332065 (all k such that k^n is a sum of distinct n-th powers).

Table[SelectFirst[Range@ 120, 2 #^n > (# + 1)^n &], {n, 0, 71}] (* Michael De Vlieger, May 01 2016, Version 10 *)

for (n=2,50, x=2; while (2*x^n<=((x+1)^n),x++); print1(x","))

a(n)=1\(1-1/2^(1/n)) \\ Charles R Greathouse IV, Oct 31 2013

apply( A078607(n)=ceil(1/if(n>1,sqrtn(2,n)-1,!n+n/2)), [0..80]) \\ M. F. Hasler, May 08 2020

a(n) = ceiling(1/(2^(1/n)-1)) for n > 1. (For n = 1 resp. 0 this gives the integer 1 resp. infinity as argument of ceiling.) [Edited by M. F. Hasler, May 08 2020]
For most n, a(n) is the nearest integer to n/log(2), but there are exceptions, including n=777451915729368.
Following formulae merged in from former A230748, M. F. Hasler, May 14 2020:
a(n) = floor(1/(1-1/2^(1/n))).
a(n) = n/log(2) + O(1). - Charles R Greathouse IV, Oct 31 2013
a(n) = floor(1/(1-x)) with x^n = 1/2: f(n) = 1/(2^(1/n)-1) is never an integer for n > 1, whence floor(f(n)+1) = ceiling(f(n)) = a(n). - M. F. Hasler, Nov 02 2013, and Gabriel Conant, May 01 2016

Edited by Dean Hickerson, Dec 17 2002

Initial terms a(0) = 1 and a(1) = 2 added by M. F. Hasler, Nov 02 2013

A332101 Least m such that m^n <= Sum_{k

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 50, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 91, 93, 94, 96, 97

Offset: 0

M. F. Hasler, Apr 14 2020

nonn

In a list (1^n, 2^n, 3^n, ...) (rows of table A051128 or A051129), a(n) is the index of the first term less than or equal to the sum of all earlier terms, cf. example.

Obviously a lower bound for any s solution to s^n = Sum_{x in S} x^n, S subset of {1, ..., s-1}, cf. A030052.

			For n = 0, m^0 > Sum_{0 < k < m} k^0 = 0 for m = 0, 1 (empty sums), but 2^0 = Sum_{0 < k < 2} k^0 = 1, so a(0) = 2.
For n = 1, 1^1 > Sum_{0 < k < 1} k^1 = 0 (empty sum) and 2^1 > Sum_{0 < k < 2} k^1 = 1, but 3^1 <= Sum_{0 < k < 3} k^1 = 1 + 2, so a(1) = 3.
To find a(n) one can add up terms in row n of the table k^n until the sum equals or exceeds the next term, whose column number k is then a(n):
  n |k: 1  2   3   4    5    6          Comment
  --+---------------------------------------------------------------
  1 |  1   2   3                  1 < 2 but 1 + 2 >= 3, so a(1) = 3.
  2 |  1   4   9  16   25         1 + 4 + 9 + 16 > 25, and a(2) = 5.
  3 |  1   8  27  64  125  216    1 + 8 + 27 + 64 + 125 > 216: a(3) = 6.

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024.

Cf. A051128, A051129.

Cf. A078607, A332097 (maximum of E(s), cf comments), A030052 (least k such that k^n = sum of distinct n-th powers), A332065 (all k such that k^n is a sum of distinct n-th powers).

Table[Block[{m = 1, s = 0}, While[m^n > s, s = s + m^n; m++]; m], {n, 0, 66}] (* Michael De Vlieger, Apr 30 2020 *)

PARI
```
apply( A332101(n,s)=for(m=1,oo, s
				
```

a(n) = round(n / log(2)) + 2. (Conjectured; verified up to 10^4, in particular for 3525/log(2) = 5085.500019... and 7844/log(2) ~ 11316.49990...)

a(n) = A078607(n) + 2 for almost all n > 1. (n = 777451915729368 might be an exception to this equality or the above one.) - M. F. Hasler, May 08 2020

A332097 Maximum of s^n - Sum_{0 < x < s} x^n.

Original entry on oeis.org

1, 1, 4, 28, 317, 4606, 84477, 1919575, 47891482, 1512466345, 48627032377, 1930020260416, 77986967769593, 3624337209819538, 178110510699972510, 9381158756438306167, 548676565488760277878, 31900481466759651567625, 2189463436999785648552851, 144075114432622269076465962

Offset: 0

M. F. Hasler, May 07 2020

nonn

For small values of s, we have Sum_{0 < x < s} x^n ~ (s-1)^n, but for s > n/log(2) + 1.5 (cf. A332101) the difference E(s) = s^n - Sum_{0 < x < s} x^n becomes negative. Just before, the difference has its maximum: We have E(s) < E(s+1) <=> 2*s^n < (s+1)^n <=> s < 1/(2^(1/n)-1), so E takes its maximum at s = A078607(n), the least integer larger than this limiting value. This appears to be almost always equal to A332101(n) - 2.

Alois P. Heinz, Table of n, a(n) for n = 0..367

Cf. A030052 (least k such that k^n = sum of distinct n-th powers).
Cf. A078607 (s for which E(s) = a(n) <=> least k such that 2*k^n > (k+1)^n).
Cf. A332065 (all k such that k^n is a sum of distinct n-th powers).
Cf. A332101 (least k such that k^n <= sum of all smaller n-th powers).

a:= proc(n) option remember; `if`(n=0, 1, (s->
      s^n-add(x^n, x=1..s-1))(ceil(1/(2^(1/n)-1))))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020

a[0] = 1; a[n_] := (s = Ceiling[1/(2^(1/n) - 1)])^n - Sum[k^n, {k, 1, s - 1}]; Array[a, 20, 0] (* Amiram Eldar, May 09 2020 *)

{apply( A332097(n,s=1\(sqrtn(2,n-!n)-1))=(s+1)^n-sum(k=1,s,k^n), [0..20])}

a(n) = s^n - Sum_{0 < x < s} x^n for s = ceiling(1/(2^(1/n)-1)) = A078607(n).

A332066 Number of positive integers whose n-th power is not the sum of distinct smaller positive n-th powers.

Original entry on oeis.org

2, 6, 9, 32, 24, 30, 41, 83, 49, 62, 71, 83

Offset: 1

M. F. Hasler, Jul 19 2020

See A332065 for the numbers whose n-th power is the sum of distinct smaller positive n-th powers. This sequence counts the positive integers not in a given row n of that table, whence the formula.

			For n = 1, only s = 1 and s = 2 are not the sum of distinct smaller positive integers (to the power n = 1), for all s >= 3 on we have s^1 = 1^1 + (s-1)^1 with 1 and s-1 distinct positive integers. Thus a(1) = #{1, 2} = 2.
For n = 2, S2 = {1, 2, 3, 4, 6, 8} is the set of all s > 0 whose square is not the sum of distinct smaller squares, while 5^2 = 4^2 + 3^2, 7^2 = 6^2 + 3^2 + 2^2, and all s^2 >= 9^2 are also the sum of distinct smaller squares. Thus a(2) = #S2 = 6.

Cf. A332065, A030052, A332096, A332097, A332098, A332066.

a(n) = lim_{k -> oo} A332065(n,k) - k.

a(n) <= A332098(n) with equality iff A030052(n) = A332098(n) + 1 <=> A030052(n) > A332098(n), which happens for n = 1, 8, 10, ... The difference A332098(n) - a(n) is the number of "solutions" s (listed in rows of A332065) strictly less than the largest "non-solution" A332098(n).

A364637 a(n) is the least k > 1 that can be represented as a sum of one or more distinct positive m-th powers for 1 <= m <= n.

Original entry on oeis.org

2, 4, 9, 881, 7809, 134067, 12939267, 2029992385, 122120396036

Offset: 1

David A. Corneth and Peter Munn, Jul 30 2023

Sprague showed that for any m, all sufficiently large integers are the sum of distinct m-th powers. A001661(m) gives the largest number not of this form, so we can use A001661 to write an upper bound for the terms here.

			a(5) = 7809 as it can be written as a sum of one or more distinct positive m-th powers for 1 <= m <= 5 as follows. 1^5 + 2^5 + 6^5 = 2^4 + 6^4 + 7^4 + 8^4 = 3^3 + 5^3 + 14^3 + 17^3 = 1^2 + 8^2 + 88^2 = 7809^1 and no number less than 7809 can be written as such.

R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.

Sequences giving solutions for related problems: A001661, A030052.

For n >= 2, a(n) <= 1 + Max_{m=2..n} A001661(m).

A332096 Irregular table where T(n,m) = min_{A subset {1..m-1}} |m^n - Sum_{x in A} x^n|, for 1 <= m <= A332098(n) = largest m for which this is nonzero.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 0, 1, 0, 1, 1, 7, 18, 28, 25, 0, 1, 8, 0, 7, 1, 1, 15, 64, 158, 271, 317, 126, 45, 17, 59, 14, 2, 15, 3, 0, 2, 1, 2, 1, 2, 2, 2, 1, 2, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 31, 210, 748, 1825, 3351, 4606, 3760, 398, 131, 299, 0, 318, 0, 8

Offset: 1

M. F. Hasler, Jul 20 2020

nonn

It is known (Sprague 1948, cf. A001661) that for any n, only a finite number of positive integers are not the sum of distinct positive n-th powers. Therefore each row is finite, their lengths are given by A332098.
The number of nonzero terms in row n is A332066(n).
The column of the first zero (exact solution m^n = Sum_{x in A} x^n) in each row is given by A030052, unless A030052(n) = A332066(n) + 1 = A332098(n) + 1.

			The table reads:
  n\ m=1   2    3    4     5     6     7     8    9   10   11  12   13
----+--------------------------------------------------------------------------
  1 |  1   1                                                  (A332098(1) = 2.)
  2 |  1   3    4    2     0     1     0     1                (A332098(2) = 8.)
  3 |  1   7   18   28    25     0     1     8    0    7    1
  4 |  1  15   64  158   271   317   126    45   17   59   14   2   15  3  0 ...
  5 |  1  31  210  748  1825  3351  4606  3760  398  131  299   0  318  0  8 ...
The first column is all ones (A000012), since {1..m-1} = {} for m = 1.
The second column is 2^n - 1 = A000225 \ {0}, since {1..m-1} = {1} for m = 2.
The third column is 3^n - 2^n - 1 = |A083321(n)| for n > 1.

R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.

Cf. A001661, A332098, A332066, A030052, A332097, A078607.

Cf. A000012, A000225, A083321.

A332096(n,m,r=0)={if(r, (m<2||r<2^(n-1)) && return(r-1); my(E, t=1); while(m^n>=r, E=m--); E=abs(r-(m+!!E)^n); for(a=2,m, if(r=m && return(min(E,r-t)); while(m>=t && E, E=min(self()(n,m-1,r-m^n),E); E && E=min(self()(n,m-=1,r),E)); E, m < n/log(2)+1.5, m^n-sum(x=1,m-1,x^n), self()(n,m-1,m^n))}

For all n and m, T(n,m) <= A332097(n) = T(n,m*) with m* = A078607(n).

For m <= m* + 1, T(n,m) = m^n - Sum_{0 < x < m} x^n.

A332098 Largest m for which m^n = Sum_{x in S} x^n has no solution S subset of {1, ..., m-1}.

Original entry on oeis.org

2, 8, 11, 44, 27, 33, 42, 83, 51, 62, 72, 83

Offset: 1

M. F. Hasler, Apr 19 2020

Row n of table A332065 lists all s for which there is some S subset of {1,...,m-1} with s^n = Sum_{x in S} x^n. This is the case for all sufficiently large s (cf. reference there). Here we give the largest integer not in this list.

Sequence A030052 lists the smallest m for which there is a solution, so a(n) >= A030052(n) - 1. We have a(9) = 51 = A030052(9) + 4, a(10) = 62 = A030052(10) - 1, a(11) = 72 = A030052(11) + 4. - M. F. Hasler, May 14 2020, edited Jul 19 2020

			For n=1, we have m^n = (m-1)^n + 1^n, so S = {1, m-1} is a solution for all m > 2, but 2^n > 1^n and therefore no solution with m = 2 = a(1).
For n=2, we have a solution to m^n = Sum_{x in S} x^n for S subset of {1,...,m-1} for all m > 8 (cf. FORMULA in A332065), but no solution with m = 8 = a(2).

Cf. A332065, A030052, A332066.

a(n) = A030052(n) - 1 or a(n) > A030052(n).

a(n) < A001661(n)^(1/n).

a(8) - a(12) from M. F. Hasler, Jul 23 2020

A332099 Square array T(n,k) = k^n - Sum_{0 < i < k} e(i)*(k-i)^n where e(i) = 1 if the partial sum up to this term would remain <= k^n, or 0 else; n, k >= 1; read by falling antidiagonals.

Original entry on oeis.org

1, 1, 1, 0, 3, 1, 0, 4, 7, 1, 0, 2, 18, 15, 1, 0, 0, 28, 64, 31, 1, 0, 1, 25, 158, 210, 63, 1, 0, 0, 0, 271, 748, 664, 127, 1, 0, 1, 1, 317, 1825, 3302, 2058, 255, 1, 0, 0, 8, 126, 3351, 10735, 14068, 6304, 511, 1, 0, 2, 0, 45, 4606, 26141, 59425, 58718, 19170, 1023, 1, 0, 0, 19, 47, 3760, 50478, 183111, 318271, 241948, 58024, 2047, 1

Offset: 1

M. F. Hasler, Apr 19 2020

To compute T(n,k), start from k^n, then subtract (progressively strictly) smaller n-th powers whenever possible.

Since we subtract the smaller n-th powers in a greedy way, T(n,k) may be nonzero even if k^n is a sum of smaller n-th powers: cf. rows of A332065 for these k.

			The square array starts
  n\k: 1   2    3     4      5      6     7     8     9    10    11    12    13
  --+----------------------------------------------------------------------------
  1 |  1   1    0     0      0      0     0     0     0     0     0     0     0
  2 |  1   3    4     2      0      1     0     1     0     2     0     2     0
  3 |  1   7   18    28     25      0     1     8     0    19    15    18     0
  4 |  1  15   64   158    271    317    126   45    47    59   191    61    285
  5 |  1  31  210   748   1825   3351   4606  3760  398   131   702     0   1229
  6 |  1  63  664  3302  10735  26141  50478 77324 84477 21595 55300 29603  2093
  (...)
Columns 1, 2, 3: A000012, A000225, |A083321|, cf. FORMULA.
We have T(2,10) = 10^2 - 9^2 - 4^2 - 1 = 2, because we first have to subtract 9^2 = 81, even though 10 is in row 2 of A332065 since 10^2 - 8^2 - 6^2 = 0.

Cf. A030052 (least k such that k^n = sum of distinct n-th powers).
Cf. A332065 (all k such that k^n is a sum of distinct n-th powers).
Cf. A332101 (least k such that k^n <= sum of all smaller n-th powers).

A332099(n,k,t=k^n)={while(k&&t,t-=(k=min(sqrtnint(t,n),k-1))^n);t}

T(n,k) > 0  for  k < A030052(n), and  T(n,k) = 0  for  k = A030052(n).
T(n,k) = k^n - Sum_{0 < m < k} m^k  for  k < A332101(n).
T(n,1) = 1 = A000012(n);  T(n,2) = 2^n - 1 = A000225(n);
T(n,3) = 3^n - 2^n - 1 = |A083321(n)|.

cp's OEIS Frontend

A001661 Largest number not the sum of distinct positive n-th powers.

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A332065 Infinite square array where row n lists the integers whose n-th power is the sum of distinct n-th powers of positive integers; read by falling antidiagonals.

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A078607 Least positive integer x such that 2*x^n > (x+1)^n.

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A332101 Least m such that m^n <= Sum_{k

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A332097 Maximum of s^n - Sum_{0 < x < s} x^n.

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Maple

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A332066 Number of positive integers whose n-th power is not the sum of distinct smaller positive n-th powers.

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A364637 a(n) is the least k > 1 that can be represented as a sum of one or more distinct positive m-th powers for 1 <= m <= n.

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A332096 Irregular table where T(n,m) = min_{A subset {1..m-1}} |m^n - Sum_{x in A} x^n|, for 1 <= m <= A332098(n) = largest m for which this is nonzero.