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A329559 MM-numbers of multiset clutters (connected weak antichains of multisets).

%I A329559 #6 Nov 18 2019 22:11:00
%S A329559 1,2,3,5,7,9,11,13,17,19,23,25,27,29,31,37,41,43,47,49,53,59,61,67,71,
%T A329559 73,79,81,83,89,91,97,101,103,107,109,113,121,125,127,131,137,139,149,
%U A329559 151,157,163,167,169,173,179,181,191,193,197,199,203,211,223,227
%N A329559 MM-numbers of multiset clutters (connected weak antichains of multisets).
%C A329559 A weak antichain of multisets is a multiset of multisets, none of which is a proper subset of any other.
%F A329559 Equals {1} followed by the intersection of A305078 and A316476.
%e A329559 The sequence of terms tother with their corresponding clutters begins:
%e A329559    1: {}              37: {{1,1,2}}            91: {{1,1},{1,2}}
%e A329559    2: {{}}            41: {{6}}                97: {{3,3}}
%e A329559    3: {{1}}           43: {{1,4}}             101: {{1,6}}
%e A329559    5: {{2}}           47: {{2,3}}             103: {{2,2,2}}
%e A329559    7: {{1,1}}         49: {{1,1},{1,1}}       107: {{1,1,4}}
%e A329559    9: {{1},{1}}       53: {{1,1,1,1}}         109: {{10}}
%e A329559   11: {{3}}           59: {{7}}               113: {{1,2,3}}
%e A329559   13: {{1,2}}         61: {{1,2,2}}           121: {{3},{3}}
%e A329559   17: {{4}}           67: {{8}}               125: {{2},{2},{2}}
%e A329559   19: {{1,1,1}}       71: {{1,1,3}}           127: {{11}}
%e A329559   23: {{2,2}}         73: {{2,4}}             131: {{1,1,1,1,1}}
%e A329559   25: {{2},{2}}       79: {{1,5}}             137: {{2,5}}
%e A329559   27: {{1},{1},{1}}   81: {{1},{1},{1},{1}}   139: {{1,7}}
%e A329559   29: {{1,3}}         83: {{9}}               149: {{3,4}}
%e A329559   31: {{5}}           89: {{1,1,1,2}}         151: {{1,1,2,2}}
%t A329559 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A329559 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 A329559 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A329559 Select[Range[100],And[stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]<=1]&]
%Y A329559 Connected numbers are A305078.
%Y A329559 Stable numbers are A316476.
%Y A329559 Clutters (of sets) are A048143.
%Y A329559 Cf. A056239, A112798, A289509, A302242, A302494, A304716, A318991, A319837, A320275, A320456, A328514, A329553, A329555.
%K A329559 nonn
%O A329559 1,2
%A A329559 _Gus Wiseman_, Nov 18 2019