A322369 - OEIS

%I #4 Dec 05 2018 07:58:48

%S 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,

%T 142,166,192,225,250,302,339,393,451,515,581,675,762,866,985,1122,

%U 1255,1441,1612,1823,2059,2318,2591,2930,3275,3668,4118,4605,5125,5749

%N Number of strict 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) = 1 through a(11) = 10 strict disconnected integer partitions:

%e   (2,1)  (3,1)  (3,2)  (5,1)    (4,3)    (5,3)    (5,4)    (7,3)      (6,5)

%e                 (4,1)  (3,2,1)  (5,2)    (7,1)    (7,2)    (9,1)      (7,4)

%e                                 (6,1)    (4,3,1)  (8,1)    (5,3,2)    (8,3)

%e                                 (4,2,1)  (5,2,1)  (4,3,2)  (5,4,1)    (9,2)

%e                                                   (5,3,1)  (6,3,1)    (10,1)

%e                                                   (6,2,1)  (7,2,1)    (5,4,2)

%e                                                            (4,3,2,1)  (6,4,1)

%e                                                                       (7,3,1)

%e                                                                       (8,2,1)

%e                                                                       (5,3,2,1)

%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],And[UnsameQ@@#,Length[zsm[#]]!=1]&]],{n,30}]

%Y Cf. A054921, A218970, A286518, A304714, A304716, A305078, A305079, A322306, A322307, A322335, A322337, A322338, A322367, A322368.

%K nonn

%O 0,6

%A _Gus Wiseman_, Dec 04 2018