A328514 - OEIS

%I #9 Nov 19 2019 06:26:33

%S 1,2,3,5,11,13,17,29,31,39,41,43,47,59,65,67,73,79,83,87,101,109,113,

%T 127,129,137,139,149,157,163,167,179,181,191,195,199,211,233,235,237,

%U 241,257,269,271,277,283,293,303,313,317,319,331,339,347,349,353,365

%N MM-numbers of connected sets of sets.

%C 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}}.

%F Intersection of A302494 and A305078.

%e The sequence all connected set of sets together with their MM-numbers begins:

%e    1: {}

%e    2: {{}}

%e    3: {{1}}

%e    5: {{2}}

%e   11: {{3}}

%e   13: {{1,2}}

%e   17: {{4}}

%e   29: {{1,3}}

%e   31: {{5}}

%e   39: {{1},{1,2}}

%e   41: {{6}}

%e   43: {{1,4}}

%e   47: {{2,3}}

%e   59: {{7}}

%e   65: {{2},{1,2}}

%e   67: {{8}}

%e   73: {{2,4}}

%e   79: {{1,5}}

%e   83: {{9}}

%e   87: {{1},{1,3}}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t 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]]]]]]]]];

%t Select[Range[1000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&]

%Y The not-necessarily-connected case is A302494.

%Y BII-numbers of connected set-systems are A326749.

%Y MM-numbers of connected sets of multisets are A328513.

%Y Cf. A005117, A007947, A056239, A112798, A286518, A302242, A302569, A304714, A305078, A305079, A327398.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 20 2019