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A317786
Matula-Goebel numbers of locally connected rooted trees.
1
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.
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[#]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 07 2018
STATUS
approved