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A304717
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Number of connected strict integer partitions of n with pairwise indivisible parts.
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14
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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
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OFFSET
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1,10
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=1..70.
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EXAMPLE
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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}}
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MATHEMATICA
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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}]
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, May 17 2018
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STATUS
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approved
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