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

1, 1, 2, 2, 4, 6, 12, 8, 28, 20, 32, 38, 112, 76, 116, 58, 352, 236, 1296, 176, 540, 288, 4448, 374, 612, 1144, 1812, 824, 16640, 1316, 59968, 612, 2336, 4528, 3208, 2924, 231168, 18320, 10632, 2168, 856960, 7132, 3334400, 3776, 11684, 74080, 12679424, 4919, 19192

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 sequence 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) = A281118(A181821(n)).

a(prime(n)) = A289501(n).

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

Example

The a(6) = 6 tree-partitions of {1,1,2}:
  (112)
  ((1)(12))
  ((2)(11))
  ((1)(1)(2))
  ((1)((1)(2)))
  ((2)((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}]]]]];
allmsptrees[m_]:=Prepend[Join@@Table[Tuples[allmsptrees/@p], {p, Select[mps[m], Length[#]>1&]}], m];
Table[Length[allmsptrees[nrmptn[n]]], {n, 20}]

Crossrefs

Cf. A000311, A001055, A196545, A281118, A281119, A305936, A318762, A318812, A318813, A318846, A318848.

Sequence in context: A103299 A278246 A195204 * A228892 A267610 A336940

Adjacent sequences: A318844 A318845 A318846 * A318848 A318849 A318850

Keyword

nonn

Author

Gus Wiseman, Sep 04 2018

Extensions

More terms from Jinyuan Wang, Jun 26 2020

Status

approved