A296131 Number of twice-factorizations of n where the first factorization is strict and the latter factorizations are constant, i.e., type (P,Q,R).

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 4, 4, 1, 5, 1, 9, 2, 2, 2, 9, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 13, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 11, 1, 2, 4, 16, 2, 5, 1, 4, 2, 5, 1, 18, 1, 2, 4, 4, 2, 5, 1, 13, 5, 2, 1, 11, 2

Offset: 1

Author

Gus Wiseman, Dec 05 2017

Keywords

• nonn

Comments

a(n) is the number of ways to choose a perfect divisor of each factor in a strict factorization of n.

Examples

			The a(24) = 8 twice-factorizations: (2)*(3)*(2*2), (2)*(3)*(4), (2)*(12), (3)*(2*2*2), (3)*(8), (2*2)*(6), (4)*(6), (24).
		

Crossrefs

Cf. A001055, A005117, A045778, A063834, A089723, A279786, A281113, A296120, A296118.

Programs

Mathematica

sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Product[DivisorSigma[0,GCD@@FactorInteger[d][[All,2]]],{d,fac}],{fac,sfs[n]}],{n,100}]

Formula

Dirichlet g.f.: Product_{n > 1}(1 + A089723(n)/n^s).