A329552 Smallest MM-number of a connected set of n sets.

1, 2, 39, 195, 5655, 62205, 2674815

Offset: 0

Author

Gus Wiseman, Nov 17 2019

Keywords

• nonn
• more

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}}.

Examples

			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}}
    62205: {{1},{2},{3},{1,2},{1,3}}
  2674815: {{1},{2},{3},{1,2},{1,3},{1,4}}
		

Crossrefs

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).

Programs

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]]]]]]]]];
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&]]}]