cp's OEIS Frontend

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

%I A318559 #7 Aug 29 2018 16:49:38
%S A318559 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,
%T A318559 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,
%U A318559 4,2,3,1,15,1,2,4,4,2,3,1,12,5,2,1,8,2,2
%N A318559 Number of combinatory separations of the multiset of prime factors of n.
%C A318559 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.
%e A318559 The a(60) = 8 combinatory separations of {2,2,3,5}:
%e A318559   {1123},
%e A318559   {1,112}, {1,123}, {11,12}, {12,12},
%e A318559   {1,1,11}, {1,1,12},
%e A318559   {1,1,1,1}.
%t A318559 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A318559 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A318559 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A318559 normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
%t A318559 Table[Length[Union[Sort/@Map[normize,mps[primeMS[n]],{2}]]],{n,100}]
%Y A318559 Cf. A007716, A056239, A112798, A255906, A265947, A269134, A317533, A317791.
%Y A318559 Cf. A318393, A318396, A318560, A318562, A318565, A318566, A318567.
%K A318559 nonn
%O A318559 1,4
%A A318559 _Gus Wiseman_, Aug 28 2018