OFFSET
1,10
COMMENTS
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.
EXAMPLE
The a(34) = 13 connected strict integer partitions with pairwise indivisible parts are (34), (18,16), (20,14), (22,12), (24,10), (26,8), (28,6), (30,4), (14,12,8), (15,10,9), (20,8,6), (14,10,6,4), (15,9,6,4). Their corresponding multiset multisystems (see A112798, A302242) are the following.
(34): {{1,7}}
(30 4): {{1,2,3},{1,1}}
(28 6): {{1,1,4},{1,2}}
(26 8): {{1,6},{1,1,1}}
(24 10): {{1,1,1,2},{1,3}}
(22 12): {{1,5},{1,1,2}}
(20 14): {{1,1,3},{1,4}}
(20 8 6): {{1,1,3},{1,1,1},{1,2}}
(18 16): {{1,2,2},{1,1,1,1}}
(15 10 9): {{2,3},{1,3},{2,2}}
(15 9 6 4): {{2,3},{2,2},{1,2},{1,1}}
(14 12 8): {{1,4},{1,1,2},{1,1,1}}
(14 10 6 4): {{1,4},{1,3},{1,2},{1,1}}
MATHEMATICA
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c==={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[zsm[#]]===1&&Select[Tuples[#, 2], UnsameQ@@#&&Divisible@@#&]==={}&]], {n, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 17 2018
STATUS
approved