OFFSET
1,4
COMMENTS
The z-density of a multiset S is defined to be Sum_{s in S} (omega(s) - 1) - omega(lcm(S)), where omega = A001221 and lcm is least common multiple.
Given a finite multiset S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. 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(18) = 8 integer partitions are (18), (14,4), (10,8), (9,9), (10,4,4), (6,4,4,4), (3,3,3,3,3,3), (2,2,2,2,2,2,2,2,2).
The a(20) = 7 integer partitions are (20), (14,6), (12,8), (10,6,4), (5,5,5,5), (4,4,4,4,4), (2,2,2,2,2,2,2,2,2,2).
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]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[IntegerPartitions[n], And[zensity[#]==-1, Length[zsm[#]]==1, Select[Tuples[#, 2], UnsameQ@@#&&Divisible@@#&]=={}]&]], {n, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 04 2018
EXTENSIONS
a(51)-a(81) from Robert Price, Sep 15 2018
STATUS
approved