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A318559 Number of combinatory separations of the multiset of prime factors of n. 7
1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 3, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 3, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 8, 1, 2, 4, 11, 2, 3, 1, 4, 2, 3, 1, 15, 1, 2, 4, 4, 2, 3, 1, 12, 5, 2, 1, 8, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A (headless) combinatory separation of a multiset m is a multiset of normal multisets {t_1,...,t_k} such that there exist multisets {s_1,...,s_k} with multiset union m and such that s_i has type t_i for each i = 1...k.

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

The a(60) = 8 combinatory separations of {2,2,3,5}:

  {1123},

  {1,112}, {1,123}, {11,12}, {12,12},

  {1,1,11}, {1,1,12},

  {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]]]];

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

normize[m_]:=m/.Rule@@@Table[{Union[m][[i]], i}, {i, Length[Union[m]]}];

Table[Length[Union[Sort/@Map[normize, mps[primeMS[n]], {2}]]], {n, 100}]

CROSSREFS

Cf. A007716, A056239, A112798, A255906, A265947, A269134, A317533, A317791.

Cf. A318393, A318396, A318560, A318562, A318565, A318566, A318567.

Sequence in context: A323438 A317141 A317791 * A326334 A218320 A305254

Adjacent sequences:  A318556 A318557 A318558 * A318560 A318561 A318562

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 28 2018

STATUS

approved

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Last modified December 11 12:33 EST 2019. Contains 329916 sequences. (Running on oeis4.)