|
%I #7 Apr 02 2019 14:38:02
%S 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 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 1,1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1
%N 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 T(n, k) > 0 for all values of k iff n is practical (A005153).
%F T(n, n) = A033630(n).
%F 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 Table begins as:
%e 1
%e 1,1,1
%e 1,0,1,1
%e 1,1,1,1,1,1,1
%e 1,0,0,0,1,1
%e 1,1,2,1,1,2,1,1,2,1,1,1
%e 1,0,0,0,0,0,1,1
%e 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%e 1,0,1,1,0,0,0,0,1,1,0,1,1
%e 1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1
%t 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 Cf. A005153, A027750, A030057, A033630, A119348, A225561, A237287, A322860.
%K nonn,tabf
%O 1,24
%A _Amiram Eldar_, Mar 29 2019
|
