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

Offset: 1

Author

Gus Wiseman, Oct 14 2021

Keywords

• nonn

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.

Examples

			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
		

Links

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

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.

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

PARI

A347709(n) = { my(rats=List([])); fordiv(n,z,my(yx=n/z); fordiv(yx, y, my(x = yx/y); if((y <= z) && (x <= y) && (x > 1), listput(rats,x*z/y)))); #Set(rats); }; \\ Antti Karttunen, Jan 29 2025

Extensions

More terms from Antti Karttunen, Jan 29 2025