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A329559
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MM-numbers of multiset clutters (connected weak antichains of multisets).
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20
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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, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 203, 211, 223, 227
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OFFSET
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1,2
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COMMENTS
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A weak antichain of multisets is a multiset of multisets, none of which is a proper subset of any other.
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LINKS
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FORMULA
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EXAMPLE
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The sequence of terms tother with their corresponding clutters begins:
1: {} 37: {{1,1,2}} 91: {{1,1},{1,2}}
2: {{}} 41: {{6}} 97: {{3,3}}
3: {{1}} 43: {{1,4}} 101: {{1,6}}
5: {{2}} 47: {{2,3}} 103: {{2,2,2}}
7: {{1,1}} 49: {{1,1},{1,1}} 107: {{1,1,4}}
9: {{1},{1}} 53: {{1,1,1,1}} 109: {{10}}
11: {{3}} 59: {{7}} 113: {{1,2,3}}
13: {{1,2}} 61: {{1,2,2}} 121: {{3},{3}}
17: {{4}} 67: {{8}} 125: {{2},{2},{2}}
19: {{1,1,1}} 71: {{1,1,3}} 127: {{11}}
23: {{2,2}} 73: {{2,4}} 131: {{1,1,1,1,1}}
25: {{2},{2}} 79: {{1,5}} 137: {{2,5}}
27: {{1},{1},{1}} 81: {{1},{1},{1},{1}} 139: {{1,7}}
29: {{1,3}} 83: {{9}} 149: {{3,4}}
31: {{5}} 89: {{1,1,1,2}} 151: {{1,1,2,2}}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
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]]]]]]]]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[100], And[stableQ[primeMS[#], Divisible], Length[zsm[primeMS[#]]]<=1]&]
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CROSSREFS
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Cf. A056239, A112798, A289509, A302242, A302494, A304716, A318991, A319837, A320275, A320456, A328514, A329553, A329555.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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