OFFSET
1,5
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.
EXAMPLE
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)
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 04 2018
EXTENSIONS
More terms from Jinyuan Wang, Jun 26 2020
STATUS
approved