A332066 -id:A332066 - cp's OEIS frontend

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.

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

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

cp's OEIS Frontend

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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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.

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

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