A358522 Least number k such that the k-th standard ordered tree has Matula-Goebel number n, i.e., A358506(k) = n.

1, 2, 3, 4, 5, 6, 9, 8, 11, 10, 17, 12, 33, 18, 19, 16, 257, 22, 129, 20, 35, 34, 1025, 24, 37, 66, 43, 36, 513, 38, 65537, 32, 67, 514, 69, 44, 2049, 258, 131, 40

Offset: 1

Author

Gus Wiseman, Nov 20 2022

Keywords

• nonn
• more

Comments

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Examples

			The terms together with their standard ordered trees begin:
    1: o
    2: (o)
    3: ((o))
    4: (oo)
    5: (((o)))
    6: ((o)o)
    9: ((oo))
    8: (ooo)
   11: ((o)(o))
   10: (((o))o)
   17: ((((o))))
   12: ((o)oo)
   33: (((o)o))
   18: ((oo)o)
   19: (((o))(o))
   16: (oooo)
  257: (((oo)))
   22: ((o)(o)o)
  129: ((ooo))
   20: (((o))oo)
   35: ((oo)(o))
   34: ((((o)))o)
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

Position of first appearance of n in A358506.

The sorted version is A358521.

A000108 counts ordered rooted trees, unordered A000081.

A214577 and A358377 rank trees with no permutations.

Cf. A001263, A014486, A061775, A127301, A196050, A206487, A358371, A358372, A358378, A358379, A358505.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
uv=Table[mgnum[srt[n]],{n,10000}];
Table[Position[uv,k][[1,1]],{k,Min@@Complement[Range[Max@@uv],uv]-1}]