A305254 Number of factorizations f of n into factors greater than 1 such that the graph of f is a forest.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2

Offset: 1

Author

Gus Wiseman, May 28 2018

Keywords

• nonn

Comments

Given a factorization f consisting of positive integers greater than one, let G(F) be a multigraph with one vertex for each factor and n edges between any two vertices with n common divisors greater than 1. For example, G(6,14,15,35) is a 4-cycle; G(6,12) is a 2-cycle because 6 and 12 have multiple common divisors. This sequence counts factorizations f such that G(f) is a forest, meaning it has no cycles. [Comment edited by Robert Munafo, Mar 24 2024]

Examples

			The a(72) = 14 factorizations:
     (72)
    (2*36)     (3*24)    (4*18)    (8*9)
   (2*2*18)   (2*3*12)   (2*4*9)  (3*3*8) (3*4*6)
   (2*2*2*9)  (2*2*3*6) (2*3*3*4)
  (2*2*2*3*3)
not counted: (2*6*6) because 6 and 6 share multiple divisors; likewise (6*12) because 6 and 12 share multiple divisors.
		

Crossrefs

Cf. A001970, A048143, A281116, A285572, A286518, A286520, A303386, A304714, A304716, A305149, A305193, A305253.

Programs

Mathematica

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[facs[n],Function[f,And@@(zensity[Select[f,Function[x,Divisible[#,x]]]]==-1&/@zsm[f])]]],{n,200}]

Extensions

Extensive clarification by Robert Munafo, Mar 22 2024