A318849 Number of orderless tree-partitions of a multiset whose multiplicities are the prime indices of n.

1, 1, 2, 2, 4, 6, 11, 8, 27, 20, 30, 38, 96, 74, 114, 58, 308, 234, 1052, 176, 509, 278, 3648, 374, 600, 1076, 1760, 814, 13003, 1306, 47006, 612, 2226, 4200, 3094, 2914, 172605, 16588, 9814, 2168, 640662, 6998, 2402388, 3698, 11496, 65936, 9082538, 4914, 17996

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 multiset of tree-partitions, one of each part of a multiset partition of m with at least two parts.

Examples

			The a(7) = 11 orderless tree-partitions of {1,1,1,1}:
  (1111)
  ((1)(111))
  ((11)(11))
  ((1)(1)(11))
  ((1)((1)(11)))
  ((11)((1)(1)))
  ((1)(1)(1)(1))
  ((1)((1)(1)(1)))
  ((1)(1)((1)(1)))
  ((1)((1)((1)(1))))
  (((1)(1))((1)(1)))
		

Crossrefs

Cf. A000311, A001055, A196545, A292504, A292505, A305936, A316655, A318762, A318812, A318813, A318847.

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}]]]]];
olmsptrees[m_]:=Prepend[Union@@Table[Sort/@Tuples[olmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Length[olmsptrees[nrmptn[n]]],{n,15}]

Formula

a(n) = A292504(A181821(n)).

a(prime(n)) = A141268(n).

a(2^n) = A005804(n).

Extensions

More terms from Jinyuan Wang, Jun 26 2020