OFFSET
1,4
COMMENTS
We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
All such factorizations have even length.
Image appears to be 1, 3, 4, 6, 11, ... , missing some numbers such as 2, 5, 7, 8, 9, ...
The case of alternating product 1, the case of alternating sum 0, and the reverse version are all counted by A001055.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
EXAMPLE
The a(2) = 1 through a(10) = 4 factorizations:
2*2 3*3 2*8 5*5 6*6 7*7 8*8 9*9 2*50
4*4 2*18 2*32 3*27 5*20
2*2*2*2 3*12 4*16 3*3*3*3 10*10
2*2*3*3 2*2*2*8 2*2*5*5
2*2*4*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]]}]];
recaltprod[q_]:=Product[q[[i]]^(-1)^i, {i, Length[q]}];
Table[Length[Select[facs[n^2], IntegerQ[recaltprod[#]]&]], {n, 100}]
PROG
CROSSREFS
The nonsquared nonreciprocal even-length version is A347438.
This is the restriction to perfect squares of A347439.
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.
A347466 counts factorizations of n^2.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 22 2021
EXTENSIONS
Data section extended up to a(96) by Antti Karttunen, Jul 28 2024
STATUS
approved