A347708 Number of distinct possible alternating products of odd-length factorizations of n.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 1, 2, 1, 4, 2, 1, 1, 5, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 2, 3, 1, 2, 1, 3, 2

Offset: 1

Author

Gus Wiseman, Oct 11 2021

Keywords

• nonn

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.

Note that it is sufficient to look at only length-1 and length-3 factorizations; cf. A347709.

Examples

			Representative factorizations for each of the a(180) = 7 alternating products:
  (2*2*3*3*5) -> 5
     (2*2*45) -> 45
     (2*3*30) -> 20
     (2*5*18) -> 36/5
     (2*9*10) -> 20/9
     (3*4*15) -> 45/4
        (180) -> 180
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

The version for partitions is A028310, reverse A347707.

Positions of 1's appear to be A037143 \ {1}.

The even-length version for n > 1 is A072670, strict A211159.

Counting only integers appears to give A293234, with evens A046951.

This is the odd-length case of A347460, reverse A038548.

The any-length version for partitions is A347461, reverse A347462.

The length-3 case is A347709.

A001055 counts factorizations (strict A045778, ordered A074206).

A056239 adds up prime indices, row sums of A112798.

A276024 counts distinct positive subset-sums of partitions.

A292886 counts knapsack factorizations, by sum A293627.

A301957 counts distinct subset-products of prime indices.

A304792 counts distinct subset-sums of partitions.

A347050 = factorizations w/ an alternating permutation, complement A347706.

A347441 counts odd-length factorizations with integer alternating product.

Cf. A002033, A103919, A108917, A119620, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.

Programs

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[Union[altprod/@Select[facs[n],OddQ[Length[#]]&]]],{n,100}]

PARI

altprod(facs) = prod(i=1,#facs,facs[i]^((-1)^(i-1)));
A347708aux(n, m=n, facs=List([])) = if(1==n, if((#facs)%2, altprod(facs), 0), my(newfacs, r, rats=List([])); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); r = A347708aux(n/d, d, newfacs); if(r, rats = concat(rats,r)))); (rats));
A347708(n) = if(1==n,0,#Set(A347708aux(n))); \\ Antti Karttunen, Jan 29 2025

Formula

Conjecture: For n > 1, a(n) = 1 + A347460(n) - A038548(n) + A072670(n).

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 29 2025