A318848 - OEIS

%I #9 Jun 26 2020 11:46:57

%S 1,1,1,1,2,3,5,4,12,9,12,17,34,29,44,26,92,90,277,68,171,93,806,144,

%T 197,309,581,269,2500,428,7578,236,631,1025,869,954,24198,3463,2402,

%U 712,75370,1957,243800,1040,3200,11705,776494,1612,4349,2358,8862,3993,2545777

%N Number of complete 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 sequence of tree-partitions, one of each part of a multiset partition of m with at least two parts. A tree-partition is complete if the leaves are all multisets of length 1.

%F a(n) = A281119(A181821(n)).

%F a(prime(n)) = A196545(n)

%F a(2^n) = A000311(n).

%e The a(12) = 17 complete tree-partitions of {1,1,2,3} with the leaves (x) replaced with just x:

%e   (1(1(23)))

%e   (1(2(13)))

%e   (1(3(12)))

%e   (2(1(13)))

%e   (2(3(11)))

%e   (3(1(12)))

%e   (3(2(11)))

%e   ((11)(23))

%e   ((12)(13))

%e   (1(123))

%e   (2(113))

%e   (3(112))

%e   (11(23))

%e   (12(13))

%e   (13(12))

%e   (23(11))

%e   (1123)

%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 allmsptrees[m_]:=Prepend[Join@@Table[Tuples[allmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];

%t Table[Length[Select[allmsptrees[nrmptn[n]],FreeQ[#,{_?AtomQ,__}]&]],{n,20}]

%Y Cf. A000311, A001055, A196545, A281118, A281119, A305936, A318762, A318812, A318813, A318846, A318847, A318849.

%K nonn

%O 1,5

%A _Gus Wiseman_, Sep 04 2018

%E More terms from _Jinyuan Wang_, Jun 26 2020