cp's OEIS Frontend

A328514 MM-numbers of connected sets of sets.

%I A328514 #9 Nov 19 2019 06:26:33
%S A328514 1,2,3,5,11,13,17,29,31,39,41,43,47,59,65,67,73,79,83,87,101,109,113,
%T A328514 127,129,137,139,149,157,163,167,179,181,191,195,199,211,233,235,237,
%U A328514 241,257,269,271,277,283,293,303,313,317,319,331,339,347,349,353,365
%N A328514 MM-numbers of connected sets of sets.
%C A328514 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 A328514 Intersection of A302494 and A305078.
%e A328514 The sequence all connected set of sets together with their MM-numbers begins:
%e A328514    1: {}
%e A328514    2: {{}}
%e A328514    3: {{1}}
%e A328514    5: {{2}}
%e A328514   11: {{3}}
%e A328514   13: {{1,2}}
%e A328514   17: {{4}}
%e A328514   29: {{1,3}}
%e A328514   31: {{5}}
%e A328514   39: {{1},{1,2}}
%e A328514   41: {{6}}
%e A328514   43: {{1,4}}
%e A328514   47: {{2,3}}
%e A328514   59: {{7}}
%e A328514   65: {{2},{1,2}}
%e A328514   67: {{8}}
%e A328514   73: {{2,4}}
%e A328514   79: {{1,5}}
%e A328514   83: {{9}}
%e A328514   87: {{1},{1,3}}
%t A328514 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A328514 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 A328514 Select[Range[1000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&]
%Y A328514 The not-necessarily-connected case is A302494.
%Y A328514 BII-numbers of connected set-systems are A326749.
%Y A328514 MM-numbers of connected sets of multisets are A328513.
%Y A328514 Cf. A005117, A007947, A056239, A112798, A286518, A302242, A302569, A304714, A305078, A305079, A327398.
%K A328514 nonn
%O A328514 1,2
%A A328514 _Gus Wiseman_, Oct 20 2019