



A329552 Smallest MM-number of a connected set of n sets. 7
1, 2, 39, 195, 5655, 62205, 2674815 (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}}.




The sequence of terms together with their corresponding systems begins:

        1: {}

        2: {{}}

       39: {{1},{1,2}}

      195: {{1},{2},{1,2}}

     5655: {{1},{2},{1,2},{1,3}}


  2674815: {{1},{2},{3},{1,2},{1,3},{1,4}}


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]]]]]]]]];

da=Select[Range[10000], SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&];

Table[da[[Position[PrimeOmega/@da, n][[1, 1]]]], {n, First[Split[Union[PrimeOmega/@da], #2==#1+1&]]}]


MM-numbers of connected set-systems are A328514.

The weight of the system with MM-number n is A302242(n).

Connected numbers are A305078.

Maximum connected divisor is A327076.

BII-numbers of connected sets of sets are A326749.

The smallest BII-number of a connected set of n sets is A329625(n).

Allowing edges to have repeated vertices gives A329553.

Requiring the edges to form an antichain gives A329555.

The smallest MM-number of a set of n nonempty sets is A329557(n).

Cf. A048143, A056239, A112798, A302494, A304714, A304716, A305079, A322389, A328513, A329554, A329556, A329558.

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

Sequence in context: A175760 A080920 A086338 * A042683 A319520 A209632

Adjacent sequences:  A329549 A329550 A329551 * A329553 A329554 A329555




Gus Wiseman, Nov 17 2019



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Last modified July 24 21:47 EDT 2021. Contains 346273 sequences. (Running on oeis4.)