A329558 Product of primes indexed by the first n squarefree numbers.

1, 2, 6, 30, 330, 4290, 72930, 2114970, 65564070, 2688126870, 115589455410, 5432704404270, 320529559851930, 21475480510079310, 1567710077235789630, 123849096101627380770, 10279474976435072603910, 1038226972619942332994910, 113166740015573714296445190, 12787841621759829715498306470

Offset: 0

Author

Gus Wiseman, Nov 17 2019

Keywords

• nonn

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}}. Then a(n) is the smallest MM-number of a set of n sets.

Examples

			The sequence of terms together with their corresponding systems begins:
        1: {}
        2: {{}}
        6: {{},{1}}
       30: {{},{1},{2}}
      330: {{},{1},{2},{3}}
     4290: {{},{1},{2},{3},{1,2}}
    72930: {{},{1},{2},{3},{1,2},{4}}
  2114970: {{},{1},{2},{3},{1,2},{4},{1,3}}
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 0..100

Crossrefs

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

MM-numbers of sets of sets are A302494.

The case without empty edges is A329557.

The case without singletons is A329556.

The case without empty edges or singletons is A329554.

The connected version is A329552.

Cf. A005117, A048672, A056239, A072639, A112798, A302242, A326031, A329555.

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

Programs

Mathematica

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

Formula

a(n > 0) = 2 * A329557(n - 1).

a(n) = Product_{i = 1..n} prime(A005117(i)).

Extensions

a(19) from Jinyuan Wang, Feb 24 2020