login
A329555
Smallest MM-number of a clutter (connected antichain) of n distinct sets.
7
1, 2, 377, 16211, 761917
OFFSET
0,2
COMMENTS
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}}.
EXAMPLE
The sequence of terms together with their corresponding systems begins:
1: {}
2: {{}}
377: {{1,2},{1,3}}
16211: {{1,2},{1,3},{1,4}}
761917: {{1,2},{1,3},{1,4},{2,3}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
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]]]]]]]]];
dae=Select[Range[100000], SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&&stableQ[primeMS[#], Divisible]&];
Table[dae[[Position[PrimeOmega/@dae, k][[1, 1]]]], {k, First[Split[Union[PrimeOmega/@dae], #2==#1+1&]]}]
CROSSREFS
Spanning cutters of distinct sets are counted by A048143.
MM-numbers of connected weak-antichains are A329559.
MM-numbers of sets of sets are A302494.
The smallest BII-number of a clutter with n edges is A329627.
Not requiring the edges to form an antichain gives A329552.
Connected numbers are A305078.
Stable numbers are A316476.
Other MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).
Sequence in context: A295174 A320445 A142532 * A171431 A280281 A225096
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 17 2019
STATUS
approved