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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 24 2018
STATUS
approved