A329553 Smallest MM-number of a connected set of n multisets.

1, 2, 21, 195, 1365, 25935, 435435

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: {{}}
      21: {{1},{1,1}}
     195: {{1},{2},{1,2}}
    1365: {{1},{2},{1,1},{1,2}}
   25935: {{1},{2},{1,1},{1,2},{1,1,1}}
  435435: {{1},{2},{1,1},{3},{1,2},{1,3}}
		

Crossrefs

MM-numbers of connected sets of sets 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 set-systems are A326749.

The smallest BII-number of a connected set-system is A329625.

The case of strict edges is A329552.

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

Cf. A056239, A112798, A302494, A305079, A329554, A329555, 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[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[#]&&Length[zsm[primeMS[#]]]<=1&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]