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%I #9 Jun 26 2020 09:15:19
%S 1,1,2,2,4,6,11,8,27,20,30,38,96,74,114,58,308,234,1052,176,509,278,
%T 3648,374,600,1076,1760,814,13003,1306,47006,612,2226,4200,3094,2914,
%U 172605,16588,9814,2168,640662,6998,2402388,3698,11496,65936,9082538,4914,17996
%N Number of orderless tree-partitions of a multiset whose multiplicities are the prime indices of n.
%C This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%C A tree-partition of m is either m itself or a multiset of tree-partitions, one of each part of a multiset partition of m with at least two parts.
%F a(n) = A292504(A181821(n)).
%F a(prime(n)) = A141268(n).
%F a(2^n) = A005804(n).
%e The a(7) = 11 orderless tree-partitions of {1,1,1,1}:
%e (1111)
%e ((1)(111))
%e ((11)(11))
%e ((1)(1)(11))
%e ((1)((1)(11)))
%e ((11)((1)(1)))
%e ((1)(1)(1)(1))
%e ((1)((1)(1)(1)))
%e ((1)(1)((1)(1)))
%e ((1)((1)((1)(1))))
%e (((1)(1))((1)(1)))
%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 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t olmsptrees[m_]:=Prepend[Union@@Table[Sort/@Tuples[olmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
%t Table[Length[olmsptrees[nrmptn[n]]],{n,15}]
%Y Cf. A000311, A001055, A196545, A292504, A292505, A305936, A316655, A318762, A318812, A318813, A318847.
%K nonn
%O 1,3
%A _Gus Wiseman_, Sep 04 2018
%E More terms from _Jinyuan Wang_, Jun 26 2020
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