A305080 - OEIS

%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