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

Author

Gus Wiseman, Aug 28 2018

Keywords

• nonn

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.

Examples

			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}.
		

Crossrefs

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

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

Programs

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}]