A329557 Smallest MM-number of a set of n nonempty sets.

1, 3, 15, 165, 2145, 36465, 1057485

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.

Formula

a(n) = A329558(n + 1)/2.

Example

The sequence of terms together with their corresponding systems begins:
        1: {}
        3: {{1}}
       15: {{1},{2}}
      165: {{1},{2},{3}}
     2145: {{1},{2},{3},{1,2}}
    36465: {{1},{2},{3},{1,2},{4}}
  1057485: {{1},{2},{3},{1,2},{4},{1,3}}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
dae=Select[Range[10000], SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#], 1]&];
Table[dae[[Position[PrimeOmega/@dae, k][[1, 1]]]], {k, First[Split[Union[PrimeOmega/@dae], #2==#1+1&]]}]

Crossrefs

MM-numbers of sets of sets are A302494.

MM-numbers of sets of nonempty sets are A329629.

The version allowing empty sets is A329558.

The version without singletons is A329554.

Cf. A056239, A072639, A112798, A302242, A326031, A329552, A329555, A329556.

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

Sequence in context: A005016 A304998 A105611 * A108975 A097489 A080696

Adjacent sequences: A329554 A329555 A329556 * A329558 A329559 A329560

Keyword

nonn,more

Author

Gus Wiseman, Nov 17 2019

Status

approved