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A318849 Number of orderless tree-partitions of a multiset whose multiplicities are the prime indices of n. 9
1, 1, 2, 2, 4, 6, 11, 8, 27, 20, 30, 38, 96, 74, 114, 58, 308, 234, 1052, 176, 509, 278, 3648, 374, 600, 1076, 1760, 814, 13003, 1306, 47006, 612, 2226, 4200, 3094, 2914, 172605, 16588, 9814, 2168, 640662, 6998, 2402388, 3698, 11496, 65936, 9082538, 4914, 17996 (list; graph; refs; listen; history; text; internal format)
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 multiset 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) = A292504(A181821(n)).

a(prime(n)) = A141268(n).

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

EXAMPLE

The a(7) = 11 orderless tree-partitions of {1,1,1,1}:

  (1111)

  ((1)(111))

  ((11)(11))

  ((1)(1)(11))

  ((1)((1)(11)))

  ((11)((1)(1)))

  ((1)(1)(1)(1))

  ((1)((1)(1)(1)))

  ((1)(1)((1)(1)))

  ((1)((1)((1)(1))))

  (((1)(1))((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}]]]]];

olmsptrees[m_]:=Prepend[Union@@Table[Sort/@Tuples[olmsptrees/@p], {p, Select[mps[m], Length[#]>1&]}], m];

Table[Length[olmsptrees[nrmptn[n]]], {n, 15}]

CROSSREFS

Cf. A000311, A001055, A196545, A292504, A292505, A305936, A316655, A318762, A318812, A318813, A318847.

Sequence in context: A032237 A276061 A216958 * A339587 A293014 A124346

Adjacent sequences:  A318846 A318847 A318848 * A318850 A318851 A318852

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 15:06 EDT 2021. Contains 346307 sequences. (Running on oeis4.)