OFFSET
1,4
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.
EXAMPLE
The a(n) alternating products for n = 1, 4, 8, 12, 24, 30, 36, 48, 60, 120:
1 4 8 12 24 30 36 48 60 120
1 2 3 6 10/3 9 12 15 30
1/2 3/4 8/3 5/6 4 16/3 20/3 40/3
1/3 2/3 3/10 1 3 15/4 15/2
3/8 2/15 4/9 3/4 12/5 24/5
1/6 1/4 1/3 3/5 10/3
1/9 3/16 5/12 5/6
1/12 4/15 8/15
3/20 3/10
1/15 5/24
2/15
3/40
1/30
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/@facs[n]]], {n, 100}]
CROSSREFS
Positions of 1's are 1 and A000040.
Positions of 2's appear to be A001358.
Positions of 3's appear to be A030078.
Dominates A038548, the version for reverse-alternating product.
Counting only integers gives A046951.
The even-length case is A072670.
The odd-length case is A347708.
The length-3 case is A347709.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 06 2021
STATUS
approved