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%I #12 Nov 18 2019 08:47:59
%S 1,2,39,195,5655,62205,2674815
%N Smallest MM-number of a connected set of n 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}}.
%e The sequence of terms together with their corresponding systems begins:
%e 1: {}
%e 2: {{}}
%e 39: {{1},{1,2}}
%e 195: {{1},{2},{1,2}}
%e 5655: {{1},{2},{1,2},{1,3}}
%e 62205: {{1},{2},{3},{1,2},{1,3}}
%e 2674815: {{1},{2},{3},{1,2},{1,3},{1,4}}
%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 da=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&];
%t Table[da[[Position[PrimeOmega/@da,n][[1,1]]]],{n,First[Split[Union[PrimeOmega/@da],#2==#1+1&]]}]
%Y MM-numbers of connected set-systems are A328514.
%Y The weight of the system with MM-number n is A302242(n).
%Y Connected numbers are A305078.
%Y Maximum connected divisor is A327076.
%Y BII-numbers of connected sets of sets are A326749.
%Y The smallest BII-number of a connected set of n sets is A329625(n).
%Y Allowing edges to have repeated vertices gives A329553.
%Y Requiring the edges to form an antichain gives A329555.
%Y The smallest MM-number of a set of n nonempty sets is A329557(n).
%Y Cf. A048143, A056239, A112798, A302494, A304714, A304716, A305079, A322389, A328513, A329554, A329556, A329558.
%Y Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Nov 17 2019
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