OFFSET
1,4
COMMENTS
A factorization of n is a weakly increasing 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
Antti Karttunen, Table of n, a(n) for n = 1..65537
PlanetMath, alternating sum
EXAMPLE
The factorizations for n = 4, 16, 36, 48, 54, 64, 108:
(4) (16) (36) (48) (54) (64) (108)
(2*2) (4*4) (6*6) (2*4*6) (2*3*9) (8*8) (2*6*9)
(2*2*4) (2*2*9) (3*4*4) (3*3*6) (2*4*8) (3*6*6)
(2*2*2*2) (2*3*6) (2*2*12) (4*4*4) (2*2*27)
(3*3*4) (2*2*2*2*3) (2*2*16) (2*3*18)
(2*2*3*3) (2*2*4*4) (3*3*12)
(2*2*2*2*4) (2*2*3*3*3)
(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], IntegerQ@*altprod]], {n, 100}]
PROG
(PARI) A347437(n, m=n, ap=1, e=0) = if(1==n, if(e%2, 1==denominator(ap), 1==numerator(ap)), sumdiv(n, d, if((d>1)&&(d<=m), A347437(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023
CROSSREFS
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The restriction to powers of 2 is A344607.
The even-length case is A347438, also the case of alternating product 1.
The reciprocal version is A347439.
Allowing any alternating product < 1 gives A347440.
The odd-length case is A347441.
The reverse version is A347442.
Allowing any alternating product >= 1 gives A347456.
The ordered version is A347463.
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 06 2021
EXTENSIONS
Data section extended up to a(108) by Antti Karttunen, Oct 22 2023
STATUS
approved