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A318847 Number of tree-partitions of a multiset whose multiplicities are the prime indices of n. 4
1, 1, 2, 2, 4, 6, 12, 8, 28, 20, 32, 38, 112, 76, 116, 58, 352, 236, 1296, 176, 540, 288, 4448, 374, 612, 1144, 1812, 824, 16640, 1316, 59968, 612, 2336, 4528, 3208, 2924, 231168, 18320, 10632, 2168, 856960, 7132, 3334400, 3776, 11684, 74080, 12679424, 4919, 19192 (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 sequence 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) = A281118(A181821(n)).

a(prime(n)) = A289501(n).

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

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

Cf. A000311, A001055, A196545, A281118, A281119, A305936, A318762, A318812, A318813, A318846, A318848.

Sequence in context: A103299 A278246 A195204 * A228892 A267610 A336940

Adjacent sequences:  A318844 A318845 A318846 * A318848 A318849 A318850

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.)