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A304118 Number of z-blobs with least common multiple n > 1. 23
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,30

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).

LINKS

Table of n, a(n) for n=1..88.

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

EXAMPLE

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}}

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}]

CROSSREFS

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

Sequence in context: A324882 A031264 A318809 * A319679 A242073 A245477

Adjacent sequences:  A304115 A304116 A304117 * A304119 A304120 A304121

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 19 2018

STATUS

approved

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Last modified November 20 20:56 EST 2019. Contains 329347 sequences. (Running on oeis4.)