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 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 (list; graph; refs; listen; history; text; internal format)
 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. LINKS 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}]&, 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 Cf. A000311, A001055, A196545, A281118, A281119, A305936, A318762, A318812, A318813, A318846, A318847, A318849. Sequence in context: A171038 A023395 A316655 * A193798 A101409 A271862 Adjacent sequences:  A318845 A318846 A318847 * A318849 A318850 A318851 KEYWORD nonn AUTHOR Gus Wiseman, Sep 04 2018 EXTENSIONS More terms from Jinyuan Wang, Jun 26 2020 STATUS approved

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Last modified July 27 15:12 EDT 2021. Contains 346307 sequences. (Running on oeis4.)