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%I #5 Dec 05 2018 07:58:34
%S 1,0,1,2,3,6,7,14,17,27,34,54,63,98,118,165,207,287,345,474,574,757,
%T 931,1212,1463,1890,2292,2898,3515,4413,5303
%N Number of disconnected or empty integer partitions of n.
%C 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).
%e The a(3) = 2 through a(9) = 27 disconnected integer partitions:
%e (21) (31) (32) (51) (43) (53) (54)
%e (111) (211) (41) (321) (52) (71) (72)
%e (1111) (221) (411) (61) (332) (81)
%e (311) (2211) (322) (431) (432)
%e (2111) (3111) (331) (521) (441)
%e (11111) (21111) (421) (611) (522)
%e (111111) (511) (3221) (531)
%e (2221) (3311) (621)
%e (3211) (4211) (711)
%e (4111) (5111) (3222)
%e (22111) (22211) (3321)
%e (31111) (32111) (4221)
%e (211111) (41111) (4311)
%e (1111111) (221111) (5211)
%e (311111) (6111)
%e (2111111) (22221)
%e (11111111) (32211)
%e (33111)
%e (42111)
%e (51111)
%e (222111)
%e (321111)
%e (411111)
%e (2211111)
%e (3111111)
%e (21111111)
%e (111111111)
%t 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]]]]]]]]];
%t Table[Length[Select[IntegerPartitions[n],Length[zsm[#]]!=1&]],{n,20}]
%Y Cf. A054921, A218970, A286518, A322335, A304714, A304716, A305078, A305079, A322306, A322307, A322337, A322338, A322368, A322369.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Dec 04 2018
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