A318848 Number of complete tree-partitions of a multiset whose multiplicities are the prime indices of n.

1, 1, 1, 1, 2, 3, 5, 4, 12, 9, 12, 17, 34, 29, 44, 26, 92, 90, 277, 68, 171, 93, 806, 144, 197, 309, 581, 269, 2500, 428, 7578, 236, 631, 1025, 869, 954, 24198, 3463, 2402, 712, 75370, 1957, 243800, 1040, 3200, 11705, 776494, 1612, 4349, 2358, 8862, 3993, 2545777

Offset: 1

Author

Gus Wiseman, Sep 04 2018

Keywords

• nonn

Comments

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}.

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.

Examples

			The a(12) = 17 complete tree-partitions of {1,1,2,3} with the leaves (x) replaced with just x:
  (1(1(23)))
  (1(2(13)))
  (1(3(12)))
  (2(1(13)))
  (2(3(11)))
  (3(1(12)))
  (3(2(11)))
  ((11)(23))
  ((12)(13))
  (1(123))
  (2(113))
  (3(112))
  (11(23))
  (12(13))
  (13(12))
  (23(11))
  (1123)
		

Crossrefs

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

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
allmsptrees[m_]:=Prepend[Join@@Table[Tuples[allmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Length[Select[allmsptrees[nrmptn[n]],FreeQ[#,{?AtomQ,_}]&]],{n,20}]

Formula

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

a(prime(n)) = A196545(n)

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

Extensions

More terms from Jinyuan Wang, Jun 26 2020