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A304714
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Number of connected strict integer partitions of n.
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47
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1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 5, 2, 5, 5, 6, 5, 10, 6, 12, 12, 13, 14, 21, 17, 23, 26, 30, 31, 46, 38, 51, 55, 61, 70, 87, 85, 102, 116, 128, 138, 171, 169, 204, 225, 245, 272, 319, 334, 383, 429, 464, 515, 593, 629, 715, 790, 861, 950, 1082
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OFFSET
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1,6
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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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EXAMPLE
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The a(19) = 6 strict integer partitions are (19), (9,6,4), (10,5,4), (10,6,3), (12,4,3), (8,6,3,2). Taking the normalized prime factors of each part (see A112798, A302242), we have the following connected multiset multisystems.
(19): {{8}}
(9,6,4): {{2,2},{1,2},{1,1}}
(10,5,4): {{1,3},{3},{1,1}}
(10,6,3): {{1,3},{1,2},{2}}
(12,4,3): {{1,1,2},{1,1},{2}}
(8,6,3,2): {{1,1,1},{1,2},{2},{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&]], {n, 60}]
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CROSSREFS
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The Heinz numbers of these partitions are given by A328513.
Cf. A000009, A003963, A048143, A054921, A076078, A259936, A281116, A285572, A285573, A286518, A286520, A302242.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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