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.

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

Author

M. F. Hasler, Apr 19 2020

Keywords

• nonn
• tabl

Comments

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.

Examples

			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.
		

Crossrefs

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).

Programs

PARI

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

Formula

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)|.