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%I #7 Dec 01 2022 08:56:02
%S 9,15,18,21,23,25,27,30,33,35,36,39,42,45,46,47,49,50,51,54,55,57,60,
%T 61,63,65,66,69,70,72,73,75,77,78,81,83,84,85,87,90,91,92,93,94,95,97,
%U 98,99,100,102,103,105,108,110,111,113,114,115,117,119,120,121
%N Matula-Goebel numbers of rooted trees with a greater number of internal (non-leaf) vertices than edge-height.
%C Edge-height (A109082) is the number of edges in the longest path from root to leaf.
%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.
%H Gus Wiseman, <a href="/A358725/a358725.png">The first 64 ordered trees with a greater number of internal vertices than edge-height.</a>
%F A342507(a(n)) > A109082(a(n)).
%e The terms together with their corresponding trees begin:
%e 9: ((o)(o))
%e 15: ((o)((o)))
%e 18: (o(o)(o))
%e 21: ((o)(oo))
%e 23: (((o)(o)))
%e 25: (((o))((o)))
%e 27: ((o)(o)(o))
%e 30: (o(o)((o)))
%e 33: ((o)(((o))))
%e 35: (((o))(oo))
%e 36: (oo(o)(o))
%e 39: ((o)(o(o)))
%e 42: (o(o)(oo))
%e 45: ((o)(o)((o)))
%e 46: (o((o)(o)))
%e 47: (((o)((o))))
%e 49: ((oo)(oo))
%e 50: (o((o))((o)))
%t MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],Count[MGTree[#],_[__],{0,Infinity}]>Depth[MGTree[#]]-2&]
%Y Complement of A209638 (the case of equality).
%Y These trees are counted by A316321.
%Y Positions of positive terms in A358724.
%Y The case of equality for node-height is A358576.
%Y A000081 counts rooted trees, ordered A000108.
%Y A034781 counts rooted trees by nodes and height, ordered A080936
%Y A055277 counts rooted trees by nodes and leaves, ordered A001263.
%Y Differences: A358580, A358724, A358726, A358729.
%Y MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
%Y MG core: A000040, A000720, A001222, A007097, A056239, A112798.
%Y Cf. A185650, A206487, A358577, A358578, A358581-A358586, A358587, A358592, A358730.
%K nonn
%O 1,1
%A _Gus Wiseman_, Nov 29 2022
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