cp's OEIS Frontend

A358521 Sorted list of positions of first appearances in the sequence of Matula-Goebel numbers of standard ordered trees (A358506).

%I A358521 #7 Nov 20 2022 18:30:21
%S A358521 1,2,3,4,5,6,8,9,10,11,12,16,17,18,19,20,22,24,32,33,34,35,36,37,38,
%T A358521 40,43,44,48,64,66,67,68,69,70,72,74,75,76,80,86,88,96,128,129,131,
%U A358521 132,133,134,136,137,138,139,140,144,147,148,150,152,160,171,172
%N A358521 Sorted list of positions of first appearances in the sequence of Matula-Goebel numbers of standard ordered trees (A358506).
%C A358521 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.
%C A358521 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.
%H A358521 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 A358521 The terms together with their standard ordered trees begin:
%e A358521    1: o
%e A358521    2: (o)
%e A358521    3: ((o))
%e A358521    4: (oo)
%e A358521    5: (((o)))
%e A358521    6: ((o)o)
%e A358521    8: (ooo)
%e A358521    9: ((oo))
%e A358521   10: (((o))o)
%e A358521   11: ((o)(o))
%e A358521   12: ((o)oo)
%e A358521   16: (oooo)
%e A358521   17: ((((o))))
%e A358521   18: ((oo)o)
%e A358521   19: (((o))(o))
%e A358521   20: (((o))oo)
%t A358521 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358521 srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t A358521 mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
%t A358521 fir[q_]:=Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&];
%t A358521 fir[Table[mgnum[srt[n]],{n,100}]]
%Y A358521 Positions of first appearances in A358506.
%Y A358521 The unsorted version is A358522.
%Y A358521 A000108 counts ordered rooted trees, unordered A000081.
%Y A358521 A214577 and A358377 rank trees with no permutations.
%Y A358521 Cf. A001263, A014486, A061775, A127301, A196050, A206487, A358371, A358372, A358378, A358379, A358505.
%K A358521 nonn
%O A358521 1,2
%A A358521 _Gus Wiseman_, Nov 20 2022