A295935 Number of twice-factorizations of n where the latter factorizations are constant, i.e., type (P,P,R).

1, 1, 1, 3, 1, 2, 1, 5, 3, 2, 1, 5, 1, 2, 2, 12, 1, 5, 1, 5, 2, 2, 1, 10, 3, 2, 5, 5, 1, 5, 1, 18, 2, 2, 2, 15, 1, 2, 2, 10, 1, 5, 1, 5, 5, 2, 1, 22, 3, 5, 2, 5, 1, 10, 2, 10, 2, 2, 1, 13, 1, 2, 5, 40, 2, 5, 1, 5, 2, 5, 1, 28, 1, 2, 5, 5, 2, 5, 1, 22, 12, 2, 1

Offset

1,4

Comments

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

Links

Table of n, a(n) for n=1..83.

Formula

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

Example

The a(24) = 10 twice-factorizations are:
(2)*(2)*(2)*(3), (2)*(3)*(2*2), (3)*(2*2*2)
(2)*(2)*(6), (2*2)*(6),
(2)*(3)*(4),
(2)*(12),
(3)*(8),
(4)*(6),
(24).

Mathematica

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

Crossrefs

Cf. A000005, A001055, A052409, A052410, A089723, A279784, A281113, A284639, A295923, A295924, A295931.

Sequence in context: A214208 A279965 A285121 * A318323 A366803 A347396

Adjacent sequences: A295932 A295933 A295934 * A295936 A295937 A295938

Keyword

nonn

Author

Gus Wiseman, Nov 29 2017

Status

approved