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A336820
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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.
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12
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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
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OFFSET
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1,6
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LINKS
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FORMULA
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A(n,k) = n-1 for n <= k+1.
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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];
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CROSSREFS
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Columns k=1-11 give: A001477(n-1), A001481, A004825, A004833, A004845, A004857, A004869, A004881, A004893, A004905, A004917.
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KEYWORD
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AUTHOR
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STATUS
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approved
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