A329556 Smallest MM-number of a set of n sets with no singletons.

1, 2, 26, 754, 32422, 1523834

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: {{}}
       26: {{},{1,2}}
      754: {{},{1,2},{1,3}}
    32422: {{},{1,2},{1,3},{1,4}}
  1523834: {{},{1,2},{1,3},{1,4},{2,3}}
		

Crossrefs

MM-numbers of sets of sets with no singletons are A329630.

The case without empty edges is A329554.

MM-numbers of sets of sets are A302494.

Cf. A056239, A112798, A302242, A329552, A329557, 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}]]]];
dae=Select[Range[100000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#],_?PrimeQ]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]