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

1,6

Links

Table of n, a(n) for n=1..36.

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.

Example

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.

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]

Crossrefs

Cf. A000043, A000396, A005153, A307223.

Sequence in context: A191419 A054373 A061847 * A309595 A329074 A296434

Adjacent sequences: A307221 A307222 A307223 * A307225 A307226 A307227

Keyword

nonn

Author

Amiram Eldar, Mar 29 2019

Status

approved