A296118 Number of ways to choose a factorization of each factor in a strict factorization of n.

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 18, 3, 3, 3, 23, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 45, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 34, 3, 12, 1, 8, 3, 12, 1, 66, 1, 3, 8, 8, 3, 12, 1, 45, 8, 3

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Formula

Dirichlet g.f.: Product_{n > 1}(1 + A001055(n)/n^s).

Example

The a(12) = 8 twice-factorizations are (2)*(2*3), (2)*(6), (3)*(2*2), (3)*(4), (2*2*3), (2*6), (3*4), (12).

Mathematica

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

Prog

(PARI)
A001055(n, m=n) = if(1==n, 1, sumdiv(n, d, if((d>1)&&(d<=m), A001055(n/d, d))));
A296118(n, m=n) = ((n<=m)*A001055(n) + sumdiv(n, d, if((d>1)&&(d<=m)&&(d<n), A001055(d)*A296118(n/d, d-1)))); \\ Antti Karttunen, Oct 08 2018

Crossrefs

Cf. A000009, A005117, A045778, A261049, A271619, A281113, A294788, A295923, A296119, A296121.

Sequence in context: A173284 A278136 A085053 * A296121 A277120 A104725

Adjacent sequences: A296115 A296116 A296117 * A296119 A296120 A296121

Keyword

nonn

Author

Gus Wiseman, Dec 05 2017

Status

approved