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A317786 Matula-Goebel numbers of locally connected rooted trees.

%I A317786 #8 Aug 08 2018 07:53:25
%S A317786 1,2,3,5,9,11,23,25,27,31,81,83,97,103,115,121,125,127,243,419,431,
%T A317786 509,515,529,563,575,625,631,661,691,709,729,961,1067,1331,1543,2095,
%U A317786 2187,2369,2575,2645,2875,2897,3001,3125,3637,3691,3803,4091,4201,4637,4663
%N A317786 Matula-Goebel numbers of locally connected rooted trees.
%C A317786 An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.
%e A317786 The sequence of locally connected trees together with their Matula-Goebel numbers begins:
%e A317786    1: o
%e A317786    2: (o)
%e A317786    3: ((o))
%e A317786    5: (((o)))
%e A317786    9: ((o)(o))
%e A317786   11: ((((o))))
%e A317786   23: (((o)(o)))
%e A317786   25: (((o))((o)))
%e A317786   27: ((o)(o)(o))
%e A317786   31: (((((o)))))
%e A317786   81: ((o)(o)(o)(o))
%e A317786   83: ((((o)(o))))
%e A317786   97: ((((o))((o))))
%t A317786 multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
%t A317786 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 A317786 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A317786 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 A317786 Select[Range[1000],rupQ[#]&]
%Y A317786 A subset of A184155.
%Y A317786 Cf. A000081, A276625, A286518, A286520, A304714, A316470, A316495, A316502, A317077, A317078, A317785.
%K A317786 nonn
%O A317786 1,2
%A A317786 _Gus Wiseman_, Aug 07 2018