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

1,21

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.

Links

Table of n, a(n) for n=1..70.

Example

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}}

Mathematica

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

Crossrefs

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

Sequence in context: A134870 A031286 A031276 * A261794 A328929 A098744

Adjacent sequences: A305077 A305078 A305079 * A305081 A305082 A305083

Keyword

nonn

Author

Gus Wiseman, May 25 2018

Status

approved