A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

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

Offset

1,24

Comments

This is also the number of distinct possible alternating products of length-3 factorizations of n, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)), and where a factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

Links

Table of n, a(n) for n=1..87.

Example

Representative factorizations for each of the a(360) = 9 alternating products:
   (2,2,90) -> 90
   (2,3,60) -> 40
   (2,4,45) -> 45/2
   (2,5,36) -> 72/5
   (2,6,30) -> 10
   (2,9,20) -> 40/9
  (2,10,18) -> 18/5
  (2,12,15) -> 5/2
   (3,8,15) -> 45/8

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], Length[#]==3&]]], {n, 100}]

Crossrefs

Allowing factorizations of any length <= 3 gives A033273.

Positions of positive terms are A033942.

Positions of 0's are A037143.

The length-2 version is A072670.

Allowing any length gives A347460, reverse A038548.

Allowing any odd length gives A347708.

A001055 counts factorizations (strict A045778, ordered A074206).

A122179 counts length-3 factorizations.

A292886 counts knapsack factorizations, by sum A293627.

A301957 counts distinct subset-products of prime indices.

A304792 counts distinct subset-sums of partitions, positive A276024.

Cf. A000040, A001358, A002033, A046951, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456, A347461.

Sequence in context: A125005 A378149 A309367 * A122179 A335877 A125203

Adjacent sequences: A347706 A347707 A347708 * A347710 A347711 A347712

Keyword

nonn

Author

Gus Wiseman, Oct 14 2021

Status

approved