A305080 Number of connected strict integer partitions of n with pairwise indivisible and squarefree parts.

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, 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, 11, 9, 13

Offset: 1

Author

Gus Wiseman, May 25 2018

Keywords

• nonn

Comments

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.

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.

Examples

			The a(52) = 6 strict partitions together with their corresponding multiset multisystems (which are clutters):
(21,15,10,6): {{2,4},{2,3},{1,3},{1,2}}
(22,14,10,6): {{1,5},{1,4},{1,3},{1,2}}
     (30,22): {{1,2,3},{1,5}}
     (38,14): {{1,8},{1,4}}
     (42,10): {{1,2,4},{1,3}}
      (46,6): {{1,9},{1,2}}
The a(60) = 8 strict partitions together with their corresponding multiset multisystems (which are clutters):
(21,15,14,10): {{2,4},{2,3},{1,4},{1,3}}
    (33,21,6): {{2,5},{2,4},{1,2}}
   (35,15,10): {{3,4},{2,3},{1,3}}
    (39,15,6): {{2,6},{2,3},{1,2}}
      (34,26): {{1,7},{1,6}}
      (38,22): {{1,8},{1,5}}
      (39,21): {{2,6},{2,4}}
      (46,14): {{1,9},{1,4}}
		

Crossrefs

Cf. A006126, A048143, A087188, A302242, A303362, A303364, A304714, A304716, A305078, A305079.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,And@@SquareFreeQ/@#,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,50}]