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A304118
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Number of z-blobs with least common multiple n > 1.
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22
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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
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OFFSET
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1,30
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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}}
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A006126, A030019, A048143, A076078, A112798, A134954, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303837, A303838.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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