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A304717 Number of connected strict integer partitions of n with pairwise indivisible parts. 14
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 2, 4, 3, 5, 2, 5, 4, 6, 3, 7, 6, 9, 5, 9, 8, 13, 10, 15, 9, 15, 13, 18, 14, 22, 21, 26, 19, 29, 24, 36, 31, 40, 35, 45, 38, 54, 55, 59, 55, 70, 69, 84, 74, 89, 86, 107, 103, 119, 115, 143, 143, 159 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,10

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. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

LINKS

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

EXAMPLE

The a(34) = 13 connected strict integer partitions with pairwise indivisible parts are (34), (18,16), (20,14), (22,12), (24,10), (26,8), (28,6), (30,4), (14,12,8), (15,10,9), (20,8,6), (14,10,6,4), (15,9,6,4). Their corresponding multiset multisystems (see A112798, A302242) are the following.

         (34): {{1,7}}

       (30 4): {{1,2,3},{1,1}}

       (28 6): {{1,1,4},{1,2}}

       (26 8): {{1,6},{1,1,1}}

      (24 10): {{1,1,1,2},{1,3}}

      (22 12): {{1,5},{1,1,2}}

      (20 14): {{1,1,3},{1,4}}

     (20 8 6): {{1,1,3},{1,1,1},{1,2}}

      (18 16): {{1,2,2},{1,1,1,1}}

    (15 10 9): {{2,3},{1,3},{2,2}}

   (15 9 6 4): {{2,3},{2,2},{1,2},{1,1}}

    (14 12 8): {{1,4},{1,1,2},{1,1,1}}

  (14 10 6 4): {{1,4},{1,3},{1,2},{1,1}}

MATHEMATICA

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c==={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[zsm[#]]===1&&Select[Tuples[#, 2], UnsameQ@@#&&Divisible@@#&]==={}&]], {n, 30}]

CROSSREFS

Cf. A000009, A003963, A006126, A048143, A051424, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A293993, A302242, A303362, A304714, A304716.

Sequence in context: A126805 A288003 A304382 * A110963 A292622 A292869

Adjacent sequences:  A304714 A304715 A304716 * A304718 A304719 A304720

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 17 2018

STATUS

approved

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Last modified April 13 10:24 EDT 2021. Contains 342935 sequences. (Running on oeis4.)