OFFSET
1,16
COMMENTS
Also the number of unordered factorizations of n with alternating sum 0.
Also the number of unordered factorizations of n with all even multiplicities.
An unordered 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 a(n) factorizations for n = 16, 64, 144, 256, 576:
4*4 8*8 12*12 16*16 24*24
2*2*2*2 2*2*4*4 2*2*6*6 2*2*8*8 3*3*8*8
2*2*2*2*2*2 3*3*4*4 4*4*4*4 4*4*6*6
2*2*2*2*3*3 2*2*2*2*4*4 2*2*12*12
2*2*2*2*2*2*2*2 2*2*2*2*6*6
2*2*3*3*4*4
2*2*2*2*2*2*3*3
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], altprod[#]==1&]], {n, 100}]
PROG
(PARI) A347438(n, m=n, k=0, t=1) = if(1==n, (1==t), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A347438(n/d, d, 1-k, t*(d^((-1)^k))))); (s)); \\ Antti Karttunen, Oct 30 2021
CROSSREFS
Positions of zeros are A000037.
Positions of nonzero terms are A000290.
The restriction to powers of 2 is A035363.
Positions of non-1's are A213367 \ {1}.
Sorted first positions are 1, 2, and all terms of A330972 squared.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
Allowing any integer alternating product gives A347437.
Allowing any integer reciprocal alternating product gives A347439.
Allowing any alternating product < 1 gives A347440.
Allowing any alternating product >= 1 gives A347456.
A046099 counts factorizations with no alternating permutations.
A344606 counts alternating permutations of prime factors.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 06 2021
EXTENSIONS
Name and comments clarified (with unordered) by Jacob Sprittulla, Oct 05 2021
STATUS
approved