A317786 Matula-Goebel numbers of locally connected rooted trees.

1, 2, 3, 5, 9, 11, 23, 25, 27, 31, 81, 83, 97, 103, 115, 121, 125, 127, 243, 419, 431, 509, 515, 529, 563, 575, 625, 631, 661, 691, 709, 729, 961, 1067, 1331, 1543, 2095, 2187, 2369, 2575, 2645, 2875, 2897, 3001, 3125, 3637, 3691, 3803, 4091, 4201, 4637, 4663

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..52.

Example

The sequence of locally connected trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   9: ((o)(o))
  11: ((((o))))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  81: ((o)(o)(o)(o))
  83: ((((o)(o))))
  97: ((((o))((o))))

Mathematica

multijoin[mss__]:=Join@@Table[Table[x, {Max[Count[#, x]&/@{mss}]}], {x, Union[mss]}];
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]]]]]]]]];
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
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]]]]];
Select[Range[1000], rupQ[#]&]

Crossrefs

A subset of A184155.

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

Sequence in context: A293036 A214125 A275466 * A019276 A097683 A334264

Adjacent sequences: A317783 A317784 A317785 * A317787 A317788 A317789

Keyword

nonn

Author

Gus Wiseman, Aug 07 2018

Status

approved