A318559 Number of combinatory separations of the multiset of prime factors of n.

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

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 A355030

Adjacent sequences: A318556 A318557 A318558 * A318560 A318561 A318562

Keyword

nonn

Author

Gus Wiseman, Aug 28 2018

Status

approved