cp's OEIS Frontend

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.

%I A332099 #13 Jan 04 2024 21:40:09
%S A332099 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,
%T A332099 0,0,271,748,664,127,1,0,1,1,317,1825,3302,2058,255,1,0,0,8,126,3351,
%U A332099 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
%N 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.
%C A332099 To compute T(n,k), start from k^n, then subtract (progressively strictly) smaller n-th powers whenever possible.
%C A332099 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.
%F A332099 T(n,k) > 0  for  k < A030052(n), and  T(n,k) = 0  for  k = A030052(n).
%F A332099 T(n,k) = k^n - Sum_{0 < m < k} m^k  for  k < A332101(n).
%F A332099 T(n,1) = 1 = A000012(n);  T(n,2) = 2^n - 1 = A000225(n);
%F A332099 T(n,3) = 3^n - 2^n - 1 = |A083321(n)|.
%e A332099 The square array starts
%e A332099   n\k: 1   2    3     4      5      6     7     8     9    10    11    12    13
%e A332099   --+----------------------------------------------------------------------------
%e A332099   1 |  1   1    0     0      0      0     0     0     0     0     0     0     0
%e A332099   2 |  1   3    4     2      0      1     0     1     0     2     0     2     0
%e A332099   3 |  1   7   18    28     25      0     1     8     0    19    15    18     0
%e A332099   4 |  1  15   64   158    271    317    126   45    47    59   191    61    285
%e A332099   5 |  1  31  210   748   1825   3351   4606  3760  398   131   702     0   1229
%e A332099   6 |  1  63  664  3302  10735  26141  50478 77324 84477 21595 55300 29603  2093
%e A332099   (...)
%e A332099 Columns 1, 2, 3: A000012, A000225, |A083321|, cf. FORMULA.
%e A332099 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.
%o A332099 (PARI) A332099(n,k,t=k^n)={while(k&&t,t-=(k=min(sqrtnint(t,n),k-1))^n);t}
%Y A332099 Cf. A030052 (least k such that k^n = sum of distinct n-th powers).
%Y A332099 Cf. A332065 (all k such that k^n is a sum of distinct n-th powers).
%Y A332099 Cf. A332101 (least k such that k^n <= sum of all smaller n-th powers).
%K A332099 nonn,tabl
%O A332099 1,5
%A A332099 _M. F. Hasler_, Apr 19 2020