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A358507 Sorted list of positions of first appearances in the sequence counting permutations of Matula-Goebel trees (A206487).

%I A358507 #7 Nov 21 2022 09:47:56
%S A358507 1,6,12,24,30,48,60,72,104,120,144,148,156,180,192,222,288,312,360,
%T A358507 390,432,444,480,576,712,720,780,832,864,900,1080,1110,1248,1260,1296,
%U A358507 1440,1560,1680,2136,2160,2262,2304,2340,2496,2520,2592,2738,2880,2886,3072
%N A358507 Sorted list of positions of first appearances in the sequence counting permutations of Matula-Goebel trees (A206487).
%C A358507 To get a permutation of a tree, we choose a permutation of the multiset of branches of each node.
%C A358507 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.
%e A358507 The terms together with their corresponding trees begin:
%e A358507     1: o
%e A358507     6: (o(o))
%e A358507    12: (oo(o))
%e A358507    24: (ooo(o))
%e A358507    30: (o(o)((o)))
%e A358507    48: (oooo(o))
%e A358507    60: (oo(o)((o)))
%e A358507    72: (ooo(o)(o))
%e A358507   104: (ooo(o(o)))
%e A358507   120: (ooo(o)((o)))
%e A358507   144: (oooo(o)(o))
%e A358507   148: (oo(oo(o)))
%e A358507   156: (oo(o)(o(o)))
%e A358507   180: (oo(o)(o)((o)))
%e A358507   192: (oooooo(o))
%e A358507   222: (o(o)(oo(o)))
%e A358507   288: (ooooo(o)(o))
%e A358507   312: (ooo(o)(o(o)))
%t A358507 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
%t A358507 MGTree[n_Integer]:=If[n===1,{},MGTree/@primeMS[n]]
%t A358507 treeperms[t_]:=Times@@Cases[t,b:{__}:>Length[Permutations[b]],{0,Infinity}];
%t A358507 fir[q_]:=Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&];
%t A358507 fir[Table[treeperms[MGTree[n]],{n,100}]]
%Y A358507 Positions of first appearances in A206487.
%Y A358507 The unsorted version is A358508.
%Y A358507 A000081 counts rooted trees, ordered A000108.
%Y A358507 A214577 and A358377 rank trees with no permutations.
%Y A358507 Cf. A001263, A032027, A061775, A127301, A196050, A358378, A358506, A358521, A358522.
%K A358507 nonn
%O A358507 1,2
%A A358507 _Gus Wiseman_, Nov 20 2022