OFFSET
0,6
COMMENTS
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).
EXAMPLE
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)
MATHEMATICA
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 04 2018
STATUS
approved