A336820 A(n,k) is the n-th number that is a sum of at most k positive k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 1, 2, 3, 5, 5, 0, 1, 2, 3, 8, 8, 6, 0, 1, 2, 3, 4, 9, 9, 7, 0, 1, 2, 3, 4, 16, 10, 10, 8, 0, 1, 2, 3, 4, 5, 17, 16, 13, 9, 0, 1, 2, 3, 4, 5, 32, 18, 17, 16, 10, 0, 1, 2, 3, 4, 5, 6, 33, 19, 24, 17, 11, 0, 1, 2, 3, 4, 5, 6, 64, 34, 32, 27, 18, 12

Offset

1,6

Links

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Formula

A(n,k) = n-1 for n <= k+1.

Example

Square array A(n,k) begins:
   0,  0,  0,  0,  0,  0,   0,   0,   0,  0,  0, ...
   1,  1,  1,  1,  1,  1,   1,   1,   1,  1,  1, ...
   2,  2,  2,  2,  2,  2,   2,   2,   2,  2,  2, ...
   3,  4,  3,  3,  3,  3,   3,   3,   3,  3,  3, ...
   4,  5,  8,  4,  4,  4,   4,   4,   4,  4,  4, ...
   5,  8,  9, 16,  5,  5,   5,   5,   5,  5,  5, ...
   6,  9, 10, 17, 32,  6,   6,   6,   6,  6,  6, ...
   7, 10, 16, 18, 33, 64,   7,   7,   7,  7,  7, ...
   8, 13, 17, 19, 34, 65, 128,   8,   8,  8,  8, ...
   9, 16, 24, 32, 35, 66, 129, 256,   9,  9,  9, ...
  10, 17, 27, 33, 36, 67, 130, 257, 512, 10, 10, ...

Maple

A:= proc() local l, w, A; l, w, A:= proc() [] end, proc() [] end,
      proc(n, k) option remember; local b; b:=
        proc(x, y) option remember; `if`(x<0 or y<1, {},
          {0, b(x, y-1)[], map(t-> t+l(k)[y], b(x-1, y))[]})
        end;
        while nops(w(k)) < n do forget(b);
          l(k):= [l(k)[], (nops(l(k))+1)^k];
          w(k):= sort([select(h-> h<l(k)[-1], b(k, nops(l(k))))[]])
        od; w(k)[n]
      end; A
    end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);

Mathematica

b[n_, k_, i_, t_] := b[n, k, i, t] = n == 0 || i > 0 && t > 0 && (b[n, k, i - 1, t] || i^k <= n && b[n - i^k, k, i, t - 1]);
A[n_, k_] := A[n, k] = Module[{m}, For[m = 1 + If[n == 1, -1, A[n - 1, k]], !b[m, k, m^(1/k) // Floor, k], m++]; m];
Table[A[n, 1+d-n], {d, 1, 14}, {n, 1, d}] // Flatten (* Jean-François Alcover, Dec 03 2020, using Alois P. Heinz's code for columns *)

Crossrefs

Columns k=1-11 give: A001477(n-1), A001481, A004825, A004833, A004845, A004857, A004869, A004881, A004893, A004905, A004917.

A(n+j,n) for j=0-3 give: A001477(n-1), A000027, A000079, A000051.

Cf. A336725.

Sequence in context: A362327 A350168 A262881 * A099173 A293377 A159880

Adjacent sequences: A336817 A336818 A336819 * A336821 A336822 A336823

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 04 2020

Status

approved