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A304716
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Number of integer partitions of n whose distinct parts are connected.
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69
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1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 3, 15, 4, 18, 12, 25, 11, 41, 17, 54, 36, 72, 44, 113, 69, 145, 113, 204, 153, 302, 220, 394, 343, 541, 475, 771, 662, 1023, 968, 1398, 1314, 1929, 1822, 2566, 2565, 3440, 3446, 4677, 4688, 6187, 6407, 8216, 8544, 10975, 11436
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OFFSET
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1,2
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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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FORMULA
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EXAMPLE
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The a(12) = 15 connected integer partitions and their corresponding connected multiset multisystems (see A112798, A302242) are the following.
(12): {{1,1,2}}
(6 6): {{1,2},{1,2}}
(8 4): {{1,1,1},{1,1}}
(9 3): {{2,2},{2}}
(10 2): {{1,3},{1}}
(4 4 4): {{1,1},{1,1},{1,1}}
(6 3 3): {{1,2},{2},{2}}
(6 4 2): {{1,2},{1,1},{1}}
(8 2 2): {{1,1,1},{1},{1}}
(3 3 3 3): {{2},{2},{2},{2}}
(4 4 2 2): {{1,1},{1,1},{1},{1}}
(6 2 2 2): {{1,2},{1},{1},{1}}
(4 2 2 2 2): {{1,1},{1},{1},{1},{1}}
(2 2 2 2 2 2): {{1},{1},{1},{1},{1},{1}}
(1 1 1 1 1 1 1 1 1 1 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], Length[zsm[Union[#]]]===1&]], {n, 30}]
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CROSSREFS
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Cf. A000009, A003963, A048143, A054921, A218970, A285572, A286518, A302242, A304714, A305078, A305079, A322306, A322307.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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