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A307223 Irregular table T(n, k) read by rows: n-th row gives number of subsets of the divisors of n which sum to k for 1 <= k <= sigma(n).

%I A307223 #15 May 02 2025 03:31:35
%S A307223 1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,1,1,1,1,2,1,1,2,1,1,2,1,1,1,1,
%T A307223 0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,1,1,0,1,
%U A307223 1,1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1
%N A307223 Irregular table T(n, k) read by rows: n-th row gives number of subsets of the divisors of n which sum to k for 1 <= k <= sigma(n).
%C A307223 T(n, k) > 0 for all values of k iff n is practical (A005153).
%H A307223 Amiram Eldar, <a href="/A307223/b307223.txt">Table of n, a(n) for n = 1..8299</a> (rows 1..100 flattened)
%F A307223 T(n, n) = A033630(n).
%F A307223 T(n, A030057(n)) = 0 if there is a 0 in the n-th row, i.e. A030057(n) <= sigma(n) or n is not practical.
%e A307223 Table begins as:
%e A307223   1
%e A307223   1,1,1
%e A307223   1,0,1,1
%e A307223   1,1,1,1,1,1,1
%e A307223   1,0,0,0,1,1
%e A307223   1,1,2,1,1,2,1,1,2,1,1,1
%e A307223   1,0,0,0,0,0,1,1
%e A307223   1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%e A307223   1,0,1,1,0,0,0,0,1,1,0,1,1
%e A307223   1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1
%t A307223 T[n_,k_] := Module[{d = Divisors[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, k}], k]]; Table[T[n, k], {n,1,10}, {k, 1, DivisorSigma[1,n]}] // Flatten
%Y A307223 Cf. A000203 (row lengths), A307224 (row products).
%Y A307223 Cf. A005153, A027750, A030057, A033630, A119348, A225561, A237287, A322860.
%K A307223 nonn,tabf
%O A307223 1,24
%A A307223 _Amiram Eldar_, Mar 29 2019