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A305080 Number of connected strict integer partitions of n with pairwise indivisible and squarefree parts. 0
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 (list; graph; refs; listen; history; text; internal format)
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 A098744 A273975

Adjacent sequences:  A305077 A305078 A305079 * A305081 A305082 A305083

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 25 2018

STATUS

approved

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Last modified September 21 19:59 EDT 2019. Contains 327282 sequences. (Running on oeis4.)