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%I #14 Oct 31 2018 21:53:37
%S 1,0,0,1,0,0,0,1,0,0,0,1,0,0,0,4,0,1,0,0,0,0,0,3,0,0,1,0,0,0,0,11,0,0,
%T 0,4,0,0,0,1
%N Number of set systems with no singletons whose multiset union is row n of A305936 (a multiset whose multiplicities are the prime indices of n).
%C A set system is a finite set of finite nonempty sets.
%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.
%e The a(36) = 4 set systems with no singletons whose multiset union is {1,1,2,2,3,4}:
%e {{1,2},{1,2,3,4}}
%e {{1,2,3},{1,2,4}}
%e {{1,2},{1,3},{2,4}}
%e {{1,2},{1,4},{2,3}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t hyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1]&];
%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t Table[Length[hyp[nrmptn[n]]],{n,30}]
%Y Cf. A000070, A000296, A000569, A050326, A056239, A112798, A283877, A292444, A305936, A306005, A318285, A318361, A320922, A320923, A320924.
%K nonn,more
%O 1,16
%A _Gus Wiseman_, Oct 29 2018
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