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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 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.)