A329554 Smallest MM-number of a set of n nonempty sets with no singletons.

1, 13, 377, 16211, 761917, 55619941, 4393975339, 443791509239, 50148440544007, 6870336354528959, 954976753279525301, 142291536238649269849, 23193520406899830985387, 3873317907952271774559629, 701070541339361191195292849, 139513037726532877047863276951

Offset: 0

Author

Gus Wiseman, Nov 17 2019

Keywords

• nonn

Comments

A multiset multisystem is a finite multiset of finite multisets. 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 multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}.

Examples

			The sequence of terms together with their corresponding systems begins:
       1: {}
      13: {{1,2}}
     377: {{1,2},{1,3}}
   16211: {{1,2},{1,3},{1,4}}
  761917: {{1,2},{1,3},{1,4},{2,3}}
		

Crossrefs

The smallest BII-number of a set of n sets is A000225(n).

BII-numbers of set-systems with no singletons are A326781.

MM-numbers of sets of nonempty sets are the odd terms of A302494.

MM-numbers of multisets of nonempty non-singleton sets are A320629.

The version with empty edges is A329556.

The version with singletons is A329557.

The version with empty edges and singletons is A329558.

Cf. A056239, A072639, A112798, A279952, A302242, A326031, A329552, A329556.

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

Programs

Mathematica

sqvs=Select[Range[2,30],SquareFreeQ[#]&&!PrimeQ[#]&];
Table[Times@@Prime/@Take[sqvs,k],{k,0,Length[sqvs]}]

Formula

a(n) = Product_{i = 1..n} prime(A120944(i)).