A329558 - OEIS

%I #15 Feb 24 2020 06:04:20

%S 1,2,6,30,330,4290,72930,2114970,65564070,2688126870,115589455410,

%T 5432704404270,320529559851930,21475480510079310,1567710077235789630,

%U 123849096101627380770,10279474976435072603910,1038226972619942332994910,113166740015573714296445190,12787841621759829715498306470

%N Product of primes indexed by the first n squarefree numbers.

%C 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.

%H Jinyuan Wang, <a href="/A329558/b329558.txt">Table of n, a(n) for n = 0..100</a>

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

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

%e The sequence of terms together with their corresponding systems begins:

%e         1: {}

%e         2: {{}}

%e         6: {{},{1}}

%e        30: {{},{1},{2}}

%e       330: {{},{1},{2},{3}}

%e      4290: {{},{1},{2},{3},{1,2}}

%e     72930: {{},{1},{2},{3},{1,2},{4}}

%e   2114970: {{},{1},{2},{3},{1,2},{4},{1,3}}

%t sqvs=Select[Range[30],SquareFreeQ];

%t Table[Times@@Prime/@Take[sqvs,k],{k,0,Length[sqvs]}]

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

%Y MM-numbers of sets of sets are A302494.

%Y The case without empty edges is A329557.

%Y The case without singletons is A329556.

%Y The case without empty edges or singletons is A329554.

%Y The connected version is A329552.

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

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

%K nonn

%O 0,2

%A _Gus Wiseman_, Nov 17 2019

%E a(19) from _Jinyuan Wang_, Feb 24 2020