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A318847 Number of tree-partitions of a multiset whose multiplicities are the prime indices of n.

%I A318847 #9 Jun 26 2020 11:47:11
%S A318847 1,1,2,2,4,6,12,8,28,20,32,38,112,76,116,58,352,236,1296,176,540,288,
%T A318847 4448,374,612,1144,1812,824,16640,1316,59968,612,2336,4528,3208,2924,
%U A318847 231168,18320,10632,2168,856960,7132,3334400,3776,11684,74080,12679424,4919,19192
%N A318847 Number of tree-partitions of a multiset whose multiplicities are the prime indices of n.
%C A318847 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 A318847 A tree-partition of m is either m itself or a sequence of tree-partitions, one of each part of a multiset partition of m with at least two parts.
%F A318847 a(n) = A281118(A181821(n)).
%F A318847 a(prime(n)) = A289501(n).
%F A318847 a(2^n) = A005804(n).
%e A318847 The a(6) = 6 tree-partitions of {1,1,2}:
%e A318847   (112)
%e A318847   ((1)(12))
%e A318847   ((2)(11))
%e A318847   ((1)(1)(2))
%e A318847   ((1)((1)(2)))
%e A318847   ((2)((1)(1)))
%t A318847 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A318847 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A318847 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A318847 allmsptrees[m_]:=Prepend[Join@@Table[Tuples[allmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
%t A318847 Table[Length[allmsptrees[nrmptn[n]]],{n,20}]
%Y A318847 Cf. A000311, A001055, A196545, A281118, A281119, A305936, A318762, A318812, A318813, A318846, A318848.
%K A318847 nonn
%O A318847 1,3
%A A318847 _Gus Wiseman_, Sep 04 2018
%E A318847 More terms from _Jinyuan Wang_, Jun 26 2020