A329555 - OEIS

%I #10 Nov 18 2019 08:48:24

%S 1,2,377,16211,761917

%N Smallest MM-number of a clutter (connected antichain) of n distinct 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      377: {{1,2},{1,3}}

%e    16211: {{1,2},{1,3},{1,4}}

%e   761917: {{1,2},{1,3},{1,4},{2,3}}

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

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];

%t dae=Select[Range[100000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&&stableQ[primeMS[#],Divisible]&];

%t Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]

%Y Spanning cutters of distinct sets are counted by A048143.

%Y MM-numbers of connected weak-antichains are A329559.

%Y MM-numbers of sets of sets are A302494.

%Y The smallest BII-number of a clutter with n edges is A329627.

%Y Not requiring the edges to form an antichain gives A329552.

%Y Connected numbers are A305078.

%Y Stable numbers are A316476.

%Y Cf. A056239, A112798, A302242, A319837, A320275, A322113, A327076, A328514, A329552, A329558, A329560, A329561.

%Y Other MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

%K nonn

%O 0,2

%A _Gus Wiseman_, Nov 17 2019