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)). The reverse-alternating product is the alternating product of the reversed sequence.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
FORMULA
a(2^n) = A000041(n).
EXAMPLE
The a(n) factorizations for n = 4, 8, 16, 32, 36, 54, 64:
(4) (8) (16) (32) (36) (54) (64)
(2*2) (2*4) (2*8) (4*8) (6*6) (3*18) (8*8)
(2*2*2) (4*4) (2*16) (2*18) (2*3*9) (2*32)
(2*2*4) (2*2*8) (3*12) (3*3*6) (4*16)
(2*2*2*2) (2*4*4) (2*2*9) (2*4*8)
(2*2*2*4) (2*3*6) (4*4*4)
(2*2*2*2*2) (3*3*4) (2*2*16)
(2*2*3*3) (2*2*2*8)
(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]]}]];
revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[facs[n], IntegerQ@*revaltprod]], {n, 100}]
PROG
(PARI) A347442(n, m=n, ap=1, e=0) = if(1==n, 1==denominator(ap), sumdiv(n, d, if((d>1)&&(d<=m), A347442(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023
CROSSREFS
Positions of 2's are A001248.
Positions of 1's are A005117.
Positions of non-1's are A013929.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The non-reverse version is A347437.
The reciprocal version is A347438.
The even-length case is A347439.
Allowing any alternating product < 1 gives A347440.
Allowing any alternating product >= 1 gives A347456.
The ordered version is A347463.
A038548 counts possible reverse-alternating products of factorizations.
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A273013 counts ordered factorizations of n^2 with alternating product 1.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 08 2021
EXTENSIONS
Data section extended up to a(108) by Antti Karttunen, Oct 22 2023
STATUS
approved