A358506 Matula-Goebel number of the n-th standard ordered rooted tree.

1, 2, 3, 4, 5, 6, 6, 8, 7, 10, 9, 12, 10, 12, 12, 16, 11, 14, 15, 20, 15, 18, 18, 24, 14, 20, 18, 24, 20, 24, 24, 32, 13, 22, 21, 28, 25, 30, 30, 40, 21, 30, 27, 36, 30, 36, 36, 48, 22, 28, 30, 40, 30, 36, 36, 48, 28, 40, 36, 48, 40, 48, 48, 64, 13, 26, 33, 44

Offset: 1

Author

Gus Wiseman, Nov 20 2022

Keywords

• nonn

Comments

First differs from A333219 at a(65) = 13, A333219(65) = 17.

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.

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.

Examples

			The first eight standard ordered trees are: o, (o), ((o)), (oo), (((o))), ((o)o), (o(o)), (ooo), with Matula-Goebel numbers: 1, 2, 3, 4, 5, 6, 6, 8.
		

Links

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

Crossrefs

For binary instead of standard encoding we have A127301.

There are exactly A206487(n) appearances of n.

For binary instead of Matula-Goebel encoding we have A358505.

Positions of first appearances are A358522, sorted A358521.

A000108 counts ordered rooted trees, unordered A000081.

A214577 and A358377 rank trees with no permutations.

Cf. A001263, A014486, A061775, A196050, A358371, A358372, A358378, A358379, A358507, A358508.

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];
Table[mgnum[srt[n]],{n,100}]