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%I #6 May 25 2018 21:59:29
%S 1,1,1,0,1,1,1,0,0,1,1,0,1,1,1,1,1,0,1,1,2,1,1,1,1,1,1,1,1,2,2,2,1,1,
%T 3,3,2,2,3,2,2,4,2,3,4,4,3,4,3,4,5,6,4,6,5,7,6,5,6,8,6,6,6,10,11,11,9,
%U 11,9,13
%N Number of connected strict integer partitions of n with pairwise indivisible and squarefree parts.
%C Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with 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.
%C Conjecture: This sequence is "eventually increasing," meaning that for all k >= 0 there exists an m >= 0 such that a(n) > k for all n > m. For k = 0 it appears we can take m = 18, for example.
%e The a(52) = 6 strict partitions together with their corresponding multiset multisystems (which are clutters):
%e (21,15,10,6): {{2,4},{2,3},{1,3},{1,2}}
%e (22,14,10,6): {{1,5},{1,4},{1,3},{1,2}}
%e (30,22): {{1,2,3},{1,5}}
%e (38,14): {{1,8},{1,4}}
%e (42,10): {{1,2,4},{1,3}}
%e (46,6): {{1,9},{1,2}}
%e The a(60) = 8 strict partitions together with their corresponding multiset multisystems (which are clutters):
%e (21,15,14,10): {{2,4},{2,3},{1,4},{1,3}}
%e (33,21,6): {{2,5},{2,4},{1,2}}
%e (35,15,10): {{3,4},{2,3},{1,3}}
%e (39,15,6): {{2,6},{2,3},{1,2}}
%e (34,26): {{1,7},{1,6}}
%e (38,22): {{1,8},{1,5}}
%e (39,21): {{2,6},{2,4}}
%e (46,14): {{1,9},{1,4}}
%t Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,And@@SquareFreeQ/@#,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,50}]
%Y Cf. A006126, A048143, A087188, A302242, A303362, A303364, A304714, A304716, A305078, A305079.
%K nonn
%O 1,21
%A _Gus Wiseman_, May 25 2018
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