A317786 - OEIS

%I #8 Aug 08 2018 07:53:25

%S 1,2,3,5,9,11,23,25,27,31,81,83,97,103,115,121,125,127,243,419,431,

%T 509,515,529,563,575,625,631,661,691,709,729,961,1067,1331,1543,2095,

%U 2187,2369,2575,2645,2875,2897,3001,3125,3637,3691,3803,4091,4201,4637,4663

%N Matula-Goebel numbers of locally connected rooted trees.

%C An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.

%e The sequence of locally connected trees together with their Matula-Goebel numbers begins:

%e    1: o

%e    2: (o)

%e    3: ((o))

%e    5: (((o)))

%e    9: ((o)(o))

%e   11: ((((o))))

%e   23: (((o)(o)))

%e   25: (((o))((o)))

%e   27: ((o)(o)(o))

%e   31: (((((o)))))

%e   81: ((o)(o)(o)(o))

%e   83: ((((o)(o))))

%e   97: ((((o))((o))))

%t multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];

%t csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], multijoin@@s[[c[[1]]]]]]]]];

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

%t rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[Length[csm[primeMS/@primeMS[n]]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];

%t Select[Range[1000],rupQ[#]&]

%Y A subset of A184155.

%Y Cf. A000081, A276625, A286518, A286520, A304714, A316470, A316495, A316502, A317077, A317078, A317785.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 07 2018