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A329555 Smallest MM-number of a clutter (connected antichain) of n distinct sets. 7
1, 2, 377, 16211, 761917 (list; graph; refs; listen; history; text; internal format)



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}}.


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


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}}


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&]]}]


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.

Cf. A056239, A112798, A302242, A319837, A320275, A322113, A327076, A328514, A329552, A329558, A329560, A329561.

Other MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

Sequence in context: A295174 A320445 A142532 * A171431 A280281 A225096

Adjacent sequences:  A329552 A329553 A329554 * A329556 A329557 A329558




Gus Wiseman, Nov 17 2019



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Last modified February 25 07:33 EST 2020. Contains 332221 sequences. (Running on oeis4.)