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A322369
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Number of strict disconnected or empty integer partitions of n.
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2
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1, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 10, 16, 17, 22, 26, 33, 36, 48, 52, 64, 76, 90, 101, 125, 142, 166, 192, 225, 250, 302, 339, 393, 451, 515, 581, 675, 762, 866, 985, 1122, 1255, 1441, 1612, 1823, 2059, 2318, 2591, 2930, 3275, 3668, 4118, 4605, 5125, 5749
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OFFSET
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0,6
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COMMENTS
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An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).
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LINKS
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EXAMPLE
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The a(3) = 1 through a(11) = 10 strict disconnected integer partitions:
(2,1) (3,1) (3,2) (5,1) (4,3) (5,3) (5,4) (7,3) (6,5)
(4,1) (3,2,1) (5,2) (7,1) (7,2) (9,1) (7,4)
(6,1) (4,3,1) (8,1) (5,3,2) (8,3)
(4,2,1) (5,2,1) (4,3,2) (5,4,1) (9,2)
(5,3,1) (6,3,1) (10,1)
(6,2,1) (7,2,1) (5,4,2)
(4,3,2,1) (6,4,1)
(7,3,1)
(8,2,1)
(5,3,2,1)
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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[Sort[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n], And[UnsameQ@@#, Length[zsm[#]]!=1]&]], {n, 30}]
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CROSSREFS
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Cf. A054921, A218970, A286518, A304714, A304716, A305078, A305079, A322306, A322307, A322335, A322337, A322338, A322367, A322368.
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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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