OFFSET
1,2
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.
The even-length case, the case of alternating product 1, and the case of alternating sum 0 are all counted by A001055.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16415
EXAMPLE
The a(2) = 2 through a(8) = 8 factorizations:
4 9 16 25 36 49 64
2*2 3*3 4*4 5*5 6*6 7*7 8*8
2*2*4 2*2*9 2*4*8
2*2*2*2 2*3*6 4*4*4
3*3*4 2*2*16
2*2*3*3 2*2*4*4
2*2*2*2*4
2*2*2*2*2*2
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[Select[facs[n^2], IntegerQ[altprod[#]]&]], {n, 100}]
PROG
CROSSREFS
The restriction to powers of 2 is A344611.
This is the restriction to perfect squares of A347437.
The nonsquared even-length version is A347438.
The additive version (partitions) is the even bisection of A347446.
The nonsquared ordered version is A347463.
The case of alternating product 1 in the ordered version is A347464.
Allowing any alternating product gives A347466.
A001055 counts factorizations.
A046099 counts factorizations with no alternating permutations.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347457 ranks partitions with integer alternating product.
Cf. A062312, A119620, A330972, A346635, A347440, A347441, A347442, A347445, A347451, A347456, A347704, A347705.
Apparently, A006881 gives the positions of 6's. - Antti Karttunen, Oct 22 2023
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 21 2021
EXTENSIONS
Data section extended up to a(92) by Antti Karttunen, Oct 22 2023
STATUS
approved