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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, 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
1,5
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.
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)|.
EXAMPLE
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.
PROG
(PARI) A332099(n, k, t=k^n)={while(k&&t, t-=(k=min(sqrtnint(t, n), k-1))^n); t}
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).
Sequence in context: A186827 A207327 A319083 * A045406 A143468 A133728
KEYWORD
nonn,tabl
AUTHOR
M. F. Hasler, Apr 19 2020
STATUS
approved