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

%I A358522 #6 Nov 20 2022 18:30:16
%S A358522 1,2,3,4,5,6,9,8,11,10,17,12,33,18,19,16,257,22,129,20,35,34,1025,24,
%T A358522 37,66,43,36,513,38,65537,32,67,514,69,44,2049,258,131,40
%N A358522 Least number k such that the k-th standard ordered tree has Matula-Goebel number n, i.e., A358506(k) = n.
%C A358522 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.
%C A358522 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.
%H A358522 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e A358522 The terms together with their standard ordered trees begin:
%e A358522     1: o
%e A358522     2: (o)
%e A358522     3: ((o))
%e A358522     4: (oo)
%e A358522     5: (((o)))
%e A358522     6: ((o)o)
%e A358522     9: ((oo))
%e A358522     8: (ooo)
%e A358522    11: ((o)(o))
%e A358522    10: (((o))o)
%e A358522    17: ((((o))))
%e A358522    12: ((o)oo)
%e A358522    33: (((o)o))
%e A358522    18: ((oo)o)
%e A358522    19: (((o))(o))
%e A358522    16: (oooo)
%e A358522   257: (((oo)))
%e A358522    22: ((o)(o)o)
%e A358522   129: ((ooo))
%e A358522    20: (((o))oo)
%e A358522    35: ((oo)(o))
%e A358522    34: ((((o)))o)
%t A358522 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358522 srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t A358522 mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
%t A358522 uv=Table[mgnum[srt[n]],{n,10000}];
%t A358522 Table[Position[uv,k][[1,1]],{k,Min@@Complement[Range[Max@@uv],uv]-1}]
%Y A358522 Position of first appearance of n in A358506.
%Y A358522 The sorted version is A358521.
%Y A358522 A000108 counts ordered rooted trees, unordered A000081.
%Y A358522 A214577 and A358377 rank trees with no permutations.
%Y A358522 Cf. A001263, A014486, A061775, A127301, A196050, A206487, A358371, A358372, A358378, A358379, A358505.
%K A358522 nonn,more
%O A358522 1,2
%A A358522 _Gus Wiseman_, Nov 20 2022