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

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

Offset

0,2

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}}.

Links

Table of n, a(n) for n=0..6.

Example

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}}

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&]]}]

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).

Sequence in context: A037630 A037756 A037644 * A365061 A110253 A185634

Adjacent sequences: A329550 A329551 A329552 * A329554 A329555 A329556

Keyword

nonn,more

Author

Gus Wiseman, Nov 17 2019

Status

approved