A322307 - OEIS

%I #5 Dec 03 2018 21:38:28

%S 0,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,2,1,2,1,2,1,3,1,1,2,2,

%T 2,2,1,2,2,2,1,3,1,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,3,1,2,2,1,2,3,1,2,

%U 2,3,1,2,1,2,2,2,2,3,1,2,1,2,1,3,2,2,2

%N Number of multisets in the swell of the n-th multiset multisystem.

%C First differs from A001221 at a(91) = 3, A001221(91) = 2.

%C The swell of a multiset partition is the set of possible joins of its connected submultisets, where the multiplicity of a vertex in the join of a set of multisets is the maximum multiplicity of the same vertex among the parts. For example the swell of {{1,1},{1,2},{2,2}} is:

%C   {1,1}

%C   {1,2}

%C   {2,2}

%C   {1,1,2}

%C   {1,2,2}

%C   {1,1,2,2}

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

%t zwell[y_]:=Union[y,Join@@Cases[Subsets[Union[y],{2}],{x_,z_}?(GCD@@#>1&):>zwell[Sort[Append[Fold[DeleteCases[#1,#2,{1},1]&,y,{x,z}],LCM[x,z]]]]]];

%t Table[Length[zwell[primeMS[n]]],{n,100}]

%Y Cf. A003963, A054921, A056239, A112798, A218970, A286518, A290103, A301957, A302242, A304716, A305078, A305079, A316556, A322306.

%K nonn

%O 1,6

%A _Gus Wiseman_, Dec 03 2018