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A303674
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Number of connected integer partitions of n > 1 whose distinct parts are pairwise indivisible and whose z-density is -1.
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0
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1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 4, 1, 6, 4, 5, 1, 8, 2, 7, 5, 11, 3, 11, 5, 13, 6, 14, 7, 19, 6, 19, 15, 24, 13, 28, 15, 33, 20, 34, 22, 46, 30, 48, 32, 57, 39, 67, 48, 76, 63, 88, 62, 104, 88, 110, 94, 130, 115, 164, 121, 172, 152, 198, 176, 229, 203, 270, 235, 293, 272, 341, 311, 375, 349, 453, 420, 506, 452, 570, 547
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OFFSET
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1,4
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COMMENTS
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The z-density of a multiset S is defined to be Sum_{s in S} (omega(s) - 1) - omega(lcm(S)), where omega = A001221 and lcm is least common multiple.
Given a finite multiset 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 multiset S is said to be connected if G(S) is a connected graph.
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LINKS
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EXAMPLE
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The a(18) = 8 integer partitions are (18), (14,4), (10,8), (9,9), (10,4,4), (6,4,4,4), (3,3,3,3,3,3), (2,2,2,2,2,2,2,2,2).
The a(20) = 7 integer partitions are (20), (14,6), (12,8), (10,6,4), (5,5,5,5), (4,4,4,4,4), (2,2,2,2,2,2,2,2,2,2).
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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[IntegerPartitions[n], And[zensity[#]==-1, Length[zsm[#]]==1, Select[Tuples[#, 2], UnsameQ@@#&&Divisible@@#&]=={}]&]], {n, 30}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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