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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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%I #52 Sep 16 2018 04:42:36

%S 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,

%T 19,15,24,13,28,15,33,20,34,22,46,30,48,32,57,39,67,48,76,63,88,62,

%U 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

%N Number of connected integer partitions of n > 1 whose distinct parts are pairwise indivisible and whose z-density is -1.

%C 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.

%C 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.

%e 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).

%e 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).

%t 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]]]]]]]]];

%t zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];

%t Table[Length[Select[IntegerPartitions[n],And[zensity[#]==-1,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,30}]

%Y Cf. A030019, A035053, A048143, A134954, A286520, A293510, A293994, A303837, A303838, A304382.

%K nonn

%O 1,4

%A _Gus Wiseman_, Jun 04 2018

%E a(51)-a(81) from _Robert Price_, Sep 15 2018