OFFSET
1,2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
EXAMPLE
The sequence all connected set of sets together with their MM-numbers begins:
1: {}
2: {{}}
3: {{1}}
5: {{2}}
11: {{3}}
13: {{1,2}}
17: {{4}}
29: {{1,3}}
31: {{5}}
39: {{1},{1,2}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
59: {{7}}
65: {{2},{1,2}}
67: {{8}}
73: {{2,4}}
79: {{1,5}}
83: {{9}}
87: {{1},{1,3}}
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[Sort[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
Select[Range[1000], SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 20 2019
STATUS
approved