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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. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

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.

LINKS

Table of n, a(n) for n=1..52.

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.

FORMULA

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.

EXAMPLE

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   ...

   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  ...

  ...| ...

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 = 14^4 + 9^4 + 8^4 + 6^4 + 4^4, 25^4 = 21^4 + 20^4 + 12^4 + 10^4 + 8^4 + 6^4 + 2^4, ...

See the link for other rows.

PROG

(PARI) 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) && (m<r^n*(r+n+1)\(n+1) || r=0), ); r), m||[m=A004736(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)

CROSSREFS

Cf. A030052 (first column), A001661.

Cf. A009003 (hypotenuse numbers; subsequence of row 2).

Cf. A001422, A001476, A046039, A194768, A194769.

Sequence in context: A330101 A330102 A061146 * A082514 A227215 A229445

Adjacent sequences:  A332062 A332063 A332064 * A332067 A332070 A332071

KEYWORD

nonn,more

AUTHOR

M. F. Hasler, Mar 31 2020

STATUS

approved

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Last modified July 12 08:23 EDT 2020. Contains 335657 sequences. (Running on oeis4.)