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

1,3

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.

Links

Table of n, a(n) for n=1..49.

Formula

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

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

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

Example

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)))

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}]

Crossrefs

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

Sequence in context: A032237 A276061 A216958 * A339587 A293014 A124346

Adjacent sequences: A318846 A318847 A318848 * A318850 A318851 A318852

Keyword

nonn

Author

Gus Wiseman, Sep 04 2018

Extensions

More terms from Jinyuan Wang, Jun 26 2020

Status

approved