A296120 Number of ways to choose a strict factorization of each factor in a strict factorization of n.

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 14, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 11, 3, 12, 1, 6, 3, 12, 1, 36, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3

Offset

1,6

Links

Table of n, a(n) for n=1..82.

Formula

Dirichlet g.f.: Product_{n > 1}(1 + A045778(n)/n^s).

Example

The a(36) = 14 twice-factorizations:
(36), (4*9), (3*12), (2*18), (2*3*6),
(4)*(9), (3)*(12), (3)*(3*4), (3)*(2*6), (2)*(18), (2)*(3*6), (2)*(2*9),
(2)*(3)*(6), (2)*(3)*(2*3).

Mathematica

sfs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sfs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
Table[Sum[Times@@Length/@sfs/@fac, {fac, sfs[n]}], {n, 100}]

Crossrefs

Cf. A000009, A005117, A045778, A050345, A279785, A281113, A294788, A296118, A296119, A296121.

Sequence in context: A294788 A254578 A348610 * A050345 A348611 A070039

Adjacent sequences: A296117 A296118 A296119 * A296121 A296122 A296123

Keyword

nonn

Author

Gus Wiseman, Dec 05 2017

Status

approved