A318948 Number of ways to choose an integer partition of each factor in a factorization of n.

1, 2, 3, 9, 7, 17, 15, 40, 39, 56, 56, 126, 101, 165, 197, 336, 297, 496, 490, 774, 837, 1114, 1255, 1948, 2007, 2638, 3127, 4123, 4565, 6201, 6842, 9131, 10311, 12904, 14988, 19516, 21637, 26995, 31488, 39250, 44583, 55418, 63261, 77683, 89935, 108068, 124754

Offset

1,2

Links

Table of n, a(n) for n=1..47.

Formula

Dirichlet g.f.: Product_{n > 1} 1 / (1 - P(n) / n^s) where P = A000041. [clarified by Ilya Gutkovskiy, Oct 26 2019]

Example

The a(4) = 9 ways: (1+1)*(1+1), (1+1+1+1), (1+1)*(2), (2)*(1+1), (2+1+1), (2)*(2), (2+2), (3+1), (4).

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]];
Table[Sum[Times@@PartitionsP/@fac, {fac, facs[n]}], {n, 10}]

Crossrefs

Cf. A000041, A001055, A001970, A063834, A065026, A066739, A066815, A121229, A281113, A284639, A318949.

Sequence in context: A244257 A171054 A205860 * A328424 A299773 A021421

Adjacent sequences: A318945 A318946 A318947 * A318949 A318950 A318951

Keyword

nonn

Author

Gus Wiseman, Sep 05 2018

Status

approved