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A303837
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Number of z-trees with least common multiple n > 1.
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26
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0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 10, 1, 1, 2, 1, 1, 4, 1, 2, 1, 4, 1, 6, 1, 1, 2, 2, 1, 4, 1, 4, 1, 1, 1, 10, 1, 1, 1
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OFFSET
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1,11
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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. Then a z-tree is a finite connected set of pairwise indivisible positive integers greater than 1 with clutter density -1.
This is a generalization to multiset systems of the usual definition of hypertree (viz. connected hypergraph F such that two distinct hyperedges of F intersect in at most a common vertex and such that every cycle of F is contained in a hyperedge).
If n is squarefree with k prime factors, then a(n) = A030019(k).
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LINKS
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EXAMPLE
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The a(72) = 6 z-trees together with the corresponding multiset systems (see A112798, A302242) are the following.
(72): {{1,1,1,2,2}}
(8,18): {{1,1,1},{1,2,2}}
(8,36): {{1,1,1},{1,1,2,2}}
(9,24): {{2,2},{1,1,1,2}}
(6,8,9): {{1,2},{1,1,1},{2,2}}
(8,9,12): {{1,1,1},{2,2},{1,1,2}}
The a(60) = 10 z-trees together with the corresponding multiset systems are the following.
(60): {{1,1,2,3}}
(4,30): {{1,1},{1,2,3}}
(6,20): {{1,2},{1,1,3}}
(10,12): {{1,3},{1,1,2}}
(12,15): {{1,1,2},{2,3}}
(12,20): {{1,1,2},{1,1,3}}
(15,20): {{2,3},{1,1,3}}
(4,6,10): {{1,1},{1,2},{1,3}}
(4,6,15): {{1,1},{1,2},{2,3}}
(4,10,15): {{1,1},{1,3},{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];
Table[Length[Select[Rest[Subsets[Rest[Divisors[n]]]], And[zensity[#]==-1, zsm[#]=={n}, Select[Tuples[#, 2], UnsameQ@@#&&Divisible@@#&]=={}]&]], {n, 2, 50}]
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CROSSREFS
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Cf. A006126, A030019, A048143, A076078, A112798, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303838, A304118.
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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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