login
A318848
Number of complete tree-partitions of a multiset whose multiplicities are the prime indices of n.
7
1, 1, 1, 1, 2, 3, 5, 4, 12, 9, 12, 17, 34, 29, 44, 26, 92, 90, 277, 68, 171, 93, 806, 144, 197, 309, 581, 269, 2500, 428, 7578, 236, 631, 1025, 869, 954, 24198, 3463, 2402, 712, 75370, 1957, 243800, 1040, 3200, 11705, 776494, 1612, 4349, 2358, 8862, 3993, 2545777
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.
FORMULA
a(n) = A281119(A181821(n)).
a(prime(n)) = A196545(n)
a(2^n) = A000311(n).
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 04 2018
EXTENSIONS
More terms from Jinyuan Wang, Jun 26 2020
STATUS
approved