A329555 - OEIS

A329555

Smallest MM-number of a clutter (connected antichain) of n distinct sets.

1, 2, 377, 16211, 761917

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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}}.

LINKS

Table of n, a(n) for n=0..4.

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.