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A303838
Number of z-forests with least common multiple n > 1.
17
0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 8, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 8, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 16, 1, 2, 3, 1, 2, 8, 1, 3, 2, 8, 1, 7, 1, 2, 3, 3, 2, 8, 1, 5, 1, 2, 1, 16, 2, 2
OFFSET
1,6
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-forest is a finite set of pairwise indivisible positive integers greater than 1 such that all connected components are z-trees, meaning they have clutter density -1.
This is a generalization to multiset systems of the usual definition of hyperforest (viz. 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) = A134954(k).
Differs from A324837 at positions {1, 180, 210, ...}. For example, a(210) = 55, A324837(210) = 49.
EXAMPLE
The a(60) = 16 z-forests together with the corresponding multiset systems (see A112798, A302242) are the following.
(60): {{1,1,2,3}}
(3,20): {{2},{1,1,3}}
(4,15): {{1,1},{2,3}}
(4,30): {{1,1},{1,2,3}}
(5,12): {{3},{1,1,2}}
(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}}
(3,4,5): {{2},{1,1},{3}}
(3,4,10): {{2},{1,1},{1,3}}
(4,5,6): {{1,1},{3},{1,2}}
(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}}
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[Rest[Subsets[Rest[Divisors[n]]]], Function[s, LCM@@s==n&&And@@Table[zensity[Select[s, Divisible[m, #]&]]==-1, {m, zsm[s]}]&&Select[Tuples[s, 2], UnsameQ@@#&&Divisible@@#&]=={}]]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 19 2018
STATUS
approved