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A322367
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Number of disconnected or empty integer partitions of n.
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3
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1, 0, 1, 2, 3, 6, 7, 14, 17, 27, 34, 54, 63, 98, 118, 165, 207, 287, 345, 474, 574, 757, 931, 1212, 1463, 1890, 2292, 2898, 3515, 4413, 5303
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OFFSET
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0,4
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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) = 2 through a(9) = 27 disconnected integer partitions:
(21) (31) (32) (51) (43) (53) (54)
(111) (211) (41) (321) (52) (71) (72)
(1111) (221) (411) (61) (332) (81)
(311) (2211) (322) (431) (432)
(2111) (3111) (331) (521) (441)
(11111) (21111) (421) (611) (522)
(111111) (511) (3221) (531)
(2221) (3311) (621)
(3211) (4211) (711)
(4111) (5111) (3222)
(22111) (22211) (3321)
(31111) (32111) (4221)
(211111) (41111) (4311)
(1111111) (221111) (5211)
(311111) (6111)
(2111111) (22221)
(11111111) (32211)
(33111)
(42111)
(51111)
(222111)
(321111)
(411111)
(2211111)
(3111111)
(21111111)
(111111111)
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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], Length[zsm[#]]!=1&]], {n, 20}]
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CROSSREFS
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Cf. A054921, A218970, A286518, A322335, A304714, A304716, A305078, A305079, A322306, A322307, A322337, A322338, A322368, A322369.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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