A301598 Number of thrice-factorizations of n.

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

Offset: 1

Author

Gus Wiseman, Mar 24 2018

Keywords

• nonn

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.

Examples

			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)).
		

Links

• 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.

Crossrefs

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

Programs

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}]

Formula

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