OFFSET
1,1
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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 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.
The clutter density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221.
EXAMPLE
4331 is the Heinz number of {18,20}, which is a z-tree corresponding to the multiset multisystem {{1,2,2},{1,1,3}}.
17927 is the Heinz number of {4,6,45}, which is a z-tree corresponding to the multiset multisystem {{1,1},{1,2},{2,2,3}}.
27391 is the Heinz number of {4,4,6,14}, which is a z-tree corresponding to the multiset multisystem {{1,1},{1,1},{1,2},{1,4}}.
MATHEMATICA
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
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]]]]]]]]];
zens[n_]:=If[n==1, 0, Total@Cases[FactorInteger[n], {p_, k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]]]];
Select[Range[300], And[zens[#]==-1, Length[zsm[primeMS[#]]]==1, Select[Tuples[primeMS[#], 2], UnsameQ@@#&&Divisible@@#&]=={}]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 25 2018
STATUS
approved