A307224 The total number of combinations for presenting the set of numbers 1 <= k <= sigma(n) as sums of distinct divisors of n.

1, 1, 0, 1, 0, 8, 0, 1, 0, 0, 0, 1088391168, 0, 0, 0, 1, 0, 2985984, 0, 2097152, 0, 0, 0, 103312130400000000000000000000000000, 0, 0, 0, 128, 0, 5888655348399321787662336000000000000, 0, 1, 0, 0, 0, 1373825949385418214640573033104853375673916456960000000000000000

Offset: 1

Author

Amiram Eldar, Mar 29 2019

Keywords

• nonn

Examples

			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.
		

Crossrefs

Cf. A000043, A000396, A005153, A307223.

Programs

Mathematica

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]

Formula

Equals Product_{k=1..sigma(n)} T(n, k), where T(n, k) is given in A307223.

a(n) > 0 iff n is practical (A005153).

a(2^k) = 1.

a(2^(p-1)*(2^p-1)) = 2^(2^p-1) for Mersenne exponents p.