OFFSET
1,8
COMMENTS
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
Note that it is sufficient to look at only length-1 and length-3 factorizations; cf. A347709.
EXAMPLE
Representative factorizations for each of the a(180) = 7 alternating products:
(2*2*3*3*5) -> 5
(2*2*45) -> 45
(2*3*30) -> 20
(2*5*18) -> 36/5
(2*9*10) -> 20/9
(3*4*15) -> 45/4
(180) -> 180
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], OddQ[Length[#]]&]]], {n, 100}]
CROSSREFS
Positions of 1's appear to be A037143 \ {1}.
The length-3 case is A347709.
A276024 counts distinct positive subset-sums of partitions.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions.
A347441 counts odd-length factorizations with integer alternating product.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 11 2021
STATUS
approved