A304118 Number of z-blobs with least common multiple n > 1.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1

Offset: 1

Author

Gus Wiseman, May 19 2018

Keywords

• nonn

Comments

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. A z-blob is a finite connected set S of pairwise indivisible positive integers greater than 1 such that no cap of S with at least two edges has clutter density -1.

If n is squarefree with k prime factors, then a(n) = A275307(k).

Examples

			The a(60) = 7 z-blobs together with the corresponding multiset systems (see A112798, A302242) are the following.
        (60): {{1,1,2,3}}
     (12,30): {{1,1,2},{1,2,3}}
     (20,30): {{1,1,3},{1,2,3}}
   (6,15,20): {{1,2},{2,3},{1,1,3}}
  (10,12,15): {{1,3},{1,1,2},{2,3}}
  (12,15,20): {{1,1,2},{2,3},{1,1,3}}
  (12,20,30): {{1,1,2},{1,1,3},{1,2,3}}
The a(120) = 14 z-blobs together with the corresponding multiset systems are the following.
       (120): {{1,1,1,2,3}}
     (24,30): {{1,1,1,2},{1,2,3}}
     (24,60): {{1,1,1,2},{1,1,2,3}}
     (30,40): {{1,2,3},{1,1,1,3}}
     (40,60): {{1,1,1,3},{1,1,2,3}}
   (6,15,40): {{1,2},{2,3},{1,1,1,3}}
  (10,15,24): {{1,3},{2,3},{1,1,1,2}}
  (12,15,40): {{1,1,2},{2,3},{1,1,1,3}}
  (12,30,40): {{1,1,2},{1,2,3},{1,1,1,3}}
  (15,20,24): {{2,3},{1,1,3},{1,1,1,2}}
  (15,24,40): {{2,3},{1,1,1,2},{1,1,1,3}}
  (20,24,30): {{1,1,3},{1,1,1,2},{1,2,3}}
  (24,30,40): {{1,1,1,2},{1,2,3},{1,1,1,3}}
  (24,40,60): {{1,1,1,2},{1,1,1,3},{1,1,2,3}}
		

Links

• Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Crossrefs

Cf. A006126, A030019, A048143, A076078, A112798, A134954, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303837, A303838.

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];
zreeQ[s_]:=And[Length[s]>=2,zensity[s]==-1];
zlobQ[s_]:=Apply[And,Composition[Not,zreeQ]/@Apply[LCM,zptns[s],{2}]];
zswell[s_]:=Union[LCM@@@Select[Subsets[s],Length[zsm[#]]==1&]];
zkernels[s_]:=Table[Select[s,Divisible[w,#]&],{w,zswell[s]}];
zptns[s_]:=Select[stableSets[zkernels[s],Length[Intersection[#1,#2]]>0&],Union@@#==s&];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[If[n==1,0,Length[Select[Rest[Subsets[Rest[Divisors[n]]]],And[zsm[#]=={n},Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={},zlobQ[#]]&]]],{n,100}]