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A329559
MM-numbers of multiset clutters (connected weak antichains of multisets).
20
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
OFFSET
1,2
COMMENTS
A weak antichain of multisets is a multiset of multisets, none of which is a proper subset of any other.
FORMULA
Equals {1} followed by the intersection of A305078 and A316476.
EXAMPLE
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}}
MATHEMATICA
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]&]
CROSSREFS
Connected numbers are A305078.
Stable numbers are A316476.
Clutters (of sets) are A048143.
Sequence in context: A316428 A277702 A279516 * A305081 A235868 A319151
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 18 2019
STATUS
approved