OFFSET
1,16
COMMENTS
An ordered factorization of n is a sequence of positive integers > 1 with product n.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
LINKS
EXAMPLE
The a(n) ordered factorizations for n = 16, 32, 36, 48, 64, 96:
4*4 8*4 6*6 12*4 8*8 24*4
8*2 16*2 12*3 24*2 16*4 48*2
2*2*2*2 2*2*4*2 18*2 2*2*6*2 32*2 3*2*8*2
4*2*2*2 2*2*3*3 3*2*4*2 2*2*4*4 4*2*6*2
2*3*3*2 4*2*3*2 2*2*8*2 6*2*4*2
3*2*2*3 6*2*2*2 2*4*4*2 8*2*3*2
3*3*2*2 4*2*2*4 12*2*2*2
4*2*4*2 2*2*12*2
4*4*2*2
8*2*2*2
2*2*2*2*2*2
MATHEMATICA
ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[Prepend[#, d]&/@ordfacs[n/d], {d, Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[ordfacs[n], EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]], {n, 100}]
PROG
(PARI) A347048(n, m=n, ap=1, e=0) = if(1==n, !(e%2) && 1==numerator(ap), sumdiv(n, d, if(d>1, A347048(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024
CROSSREFS
Positions of 0's are A005117 \ {2}.
The restriction to powers of 2 is A027306.
Positions of 3's appear to be A030514.
Positions of 1's are 1 and A082293.
The odd-length version is A347049.
Allowing any length gives A347463.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 10 2021
EXTENSIONS
Data section extended up to a(105) by Antti Karttunen, Jul 28 2024
STATUS
approved