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.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 04 2018
EXTENSIONS
More terms from Jinyuan Wang, Jun 26 2020
STATUS
approved