cp's OEIS Frontend

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

%I A329557 #10 Feb 23 2025 11:16:33
%S A329557 1,3,15,165,2145,36465,1057485,32782035,1344063435,57794727705,
%T A329557 2716352202135,160264779925965,10737740255039655,783855038617894815,
%U A329557 61924548050813690385,5139737488217536301955,519113486309971166497455,56583370007786857148222595,6393920810879914857749153235
%N A329557 Smallest MM-number of a set of n nonempty sets.
%C A329557 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}}.
%F A329557 a(n) = A329558(n + 1)/2.
%e A329557 The sequence of terms together with their corresponding systems begins:
%e A329557         1: {}
%e A329557         3: {{1}}
%e A329557        15: {{1},{2}}
%e A329557       165: {{1},{2},{3}}
%e A329557      2145: {{1},{2},{3},{1,2}}
%e A329557     36465: {{1},{2},{3},{1,2},{4}}
%e A329557   1057485: {{1},{2},{3},{1,2},{4},{1,3}}
%t A329557 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A329557 dae=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#],1]&];
%t A329557 Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]
%o A329557 (PARI) a(n) = my(k=1); prod(i=1, n, until(issquarefree(k), k++); prime(k)); \\ _Jinyuan Wang_, Feb 23 2025
%Y A329557 MM-numbers of sets of sets are A302494.
%Y A329557 MM-numbers of sets of nonempty sets are A329629.
%Y A329557 The version allowing empty sets is A329558.
%Y A329557 The version without singletons is A329554.
%Y A329557 Cf. A056239, A072639, A112798, A302242, A326031, A329552, A329555, A329556.
%Y A329557 Other MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).
%K A329557 nonn
%O A329557 0,2
%A A329557 _Gus Wiseman_, Nov 17 2019
%E A329557 More terms from _Jinyuan Wang_, Feb 23 2025