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A301598 Number of thrice-factorizations of n. 3
1, 1, 1, 4, 1, 4, 1, 10, 4, 4, 1, 16, 1, 4, 4, 34, 1, 16, 1, 16, 4, 4, 1, 54, 4, 4, 10, 16, 1, 22, 1, 80, 4, 4, 4, 78, 1, 4, 4, 54, 1, 22, 1, 16, 16, 4, 1, 181, 4, 16, 4, 16, 1, 54, 4, 54, 4, 4, 1, 102, 1, 4, 16, 254, 4, 22, 1, 16, 4, 22, 1, 272, 1, 4, 16, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

A thrice-factorization of n is a choice of a twice-factorization of each factor in a factorization of n. Thrice-factorizations correspond to intervals in the lattice form of the multiorder of integer factorizations.

LINKS

Table of n, a(n) for n=1..76.

Gus Wiseman, The (2*2*3*3) component of the lattice form of the multiorder of integer factorizations.

Gus Wiseman, The (2*2*3*3) component, version 2.

FORMULA

Dirichlet g.f.: Product_{n > 1} 1/(1 - A281113(n)/n^s).

EXAMPLE

The a(12) = 16 thrice-factorizations:

((2))*((2))*((3)), ((2))*((2)*(3)), ((3))*((2)*(2)), ((2)*(2)*(3)),

((2))*((2*3)), ((2)*(2*3)),

((2))*((6)), ((2)*(6)),

((3))*((2*2)), ((3)*(2*2)),

((3))*((4)), ((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]]}]];

twifacs[n_]:=Join@@Table[Tuples[facs/@f], {f, facs[n]}];

thrifacs[n_]:=Join@@Table[Tuples[twifacs/@f], {f, facs[n]}];

Table[Length[thrifacs[n]], {n, 15}]

CROSSREFS

Cf. A001055, A007716, A050336, A050338, A063834, A162247, A269134, A281113, A281116, A301595, A301598, A301706.

Sequence in context: A097936 A277027 A050338 * A077088 A297420 A156896

Adjacent sequences:  A301595 A301596 A301597 * A301599 A301600 A301601

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 24 2018

STATUS

approved

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Last modified June 25 21:43 EDT 2019. Contains 324357 sequences. (Running on oeis4.)