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%I #6 Nov 21 2022 09:48:02
%S 1,6,12,24,48,30,192,104,148,72,3072,60,12288,832,144,712,196608,222,
%T 786432,120,288,13312
%N Least Matula-Goebel number of a tree with exactly n permutations.
%C 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 To get a permutation of a tree, we choose a permutation of the multiset of branches of each node.
%e The terms together with their corresponding trees begin:
%e 1: o
%e 6: (o(o))
%e 12: (oo(o))
%e 24: (ooo(o))
%e 48: (oooo(o))
%e 30: (o(o)((o)))
%e 192: (oooooo(o))
%e 104: (ooo(o(o)))
%e 148: (oo(oo(o)))
%e 72: (ooo(o)(o))
%e 3072: (oooooooooo(o))
%e 60: (oo(o)((o)))
%e 12288: (oooooooooooo(o))
%e 832: (oooooo(o(o)))
%e 144: (oooo(o)(o))
%e 712: (ooo(ooo(o)))
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
%t MGTree[n_Integer]:=If[n===1,{},MGTree/@primeMS[n]]
%t treeperms[t_]:=Times @@ Cases[t,b:{__}:>Length[Permutations[b]],{0,Infinity}];
%t uv=Table[treeperms[MGTree[n]],{n,100000}];
%t Table[Position[uv,k][[1,1]],{k,Min@@Complement[Range[Max@@uv],uv]-1}]
%Y Position of first appearance of n in A206487.
%Y The sorted version is A358507.
%Y A000081 counts rooted trees, ordered A000108.
%Y A214577 and A358377 rank trees with no permutations.
%Y Cf. A001263, A032027, A061775, A127301, A196050, A358378, A358506, A358521, A358522.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, Nov 20 2022
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