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%I #8 Apr 02 2019 14:37:59
%S 1,1,0,1,0,8,0,1,0,0,0,1088391168,0,0,0,1,0,2985984,0,2097152,0,0,0,
%T 103312130400000000000000000000000000,0,0,0,128,0,
%U 5888655348399321787662336000000000000,0,1,0,0,0,1373825949385418214640573033104853375673916456960000000000000000
%N The total number of combinations for presenting the set of numbers 1 <= k <= sigma(n) as sums of distinct divisors of n.
%F Equals Product_{k=1..sigma(n)} T(n, k), where T(n, k) is given in A307223.
%F a(n) > 0 iff n is practical (A005153).
%F a(2^k) = 1.
%F a(2^(p-1)*(2^p-1)) = 2^(2^p-1) for Mersenne exponents p.
%e a(6) = 8 since the divisors of 6 are {1, 2, 3, 6}, k = 3, 6, and 9 are each the sum of 2 subsets (3: {1,2} and {3}, 6: {1,2,3} and {6}, 9: {1,2,6} and {3,6}) and the other values of k are sums of a single subset. Thus, a(6) = 1*1*2*1*1*2*1*1*2*1*1*1 = 8.
%t T[n_,k_] := Module[{d = Divisors[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, k}], k]]; a[n_] := Product[T[n, k], {k,1,DivisorSigma[1,n]}]; Array[a, 50]
%Y Cf. A000043, A000396, A005153, A307223.
%K nonn
%O 1,6
%A _Amiram Eldar_, Mar 29 2019
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