OFFSET
1,24
COMMENTS
This is also the number of distinct possible alternating products of length-3 factorizations of n, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)), and where a factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
EXAMPLE
Representative factorizations for each of the a(360) = 9 alternating products:
(2,2,90) -> 90
(2,3,60) -> 40
(2,4,45) -> 45/2
(2,5,36) -> 72/5
(2,6,30) -> 10
(2,9,20) -> 40/9
(2,10,18) -> 18/5
(2,12,15) -> 5/2
(3,8,15) -> 45/8
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Union[altprod/@Select[facs[n], Length[#]==3&]]], {n, 100}]
CROSSREFS
Allowing factorizations of any length <= 3 gives A033273.
Positions of positive terms are A033942.
Positions of 0's are A037143.
The length-2 version is A072670.
Allowing any odd length gives A347708.
A122179 counts length-3 factorizations.
A301957 counts distinct subset-products of prime indices.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 14 2021
STATUS
approved