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

1,4

Comments

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

Links

Table of n, a(n) for n=1..85.

Formula

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

Example

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).

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}]

Crossrefs

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

Sequence in context: A072342 A257089 A328892 * A345344 A319004 A234541

Adjacent sequences: A296128 A296129 A296130 * A296132 A296133 A296134

Keyword

nonn

Author

Gus Wiseman, Dec 05 2017

Status

approved