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A358576 Matula-Goebel numbers of rooted trees whose node-height equals their number of internal (non-leaf) nodes.

%I A358576 #13 Dec 01 2022 17:39:43
%S A358576 9,15,18,21,23,30,33,35,36,39,42,46,47,49,51,57,60,61,66,70,72,73,77,
%T A358576 78,83,84,87,91,92,93,94,95,98,102,111,113,114,119,120,122,123,129,
%U A358576 132,133,137,140,144,146,149,151,154,156,159,166,167,168,174,177,181
%N A358576 Matula-Goebel numbers of rooted trees whose node-height equals their number of internal (non-leaf) nodes.
%C A358576 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 A358576 Node-height is the number of nodes in the longest path from root to leaf.
%H A358576 Gus Wiseman, <a href="/A358576/a358576.png">The first 64 rooted trees whose node-height equals their number of internal nodes.</a>
%F A358576 A358552(a(n)) = A342507(a(n)).
%e A358576 The terms together with their corresponding rooted trees begin:
%e A358576    9: ((o)(o))
%e A358576   15: ((o)((o)))
%e A358576   18: (o(o)(o))
%e A358576   21: ((o)(oo))
%e A358576   23: (((o)(o)))
%e A358576   30: (o(o)((o)))
%e A358576   33: ((o)(((o))))
%e A358576   35: (((o))(oo))
%e A358576   36: (oo(o)(o))
%e A358576   39: ((o)(o(o)))
%e A358576   42: (o(o)(oo))
%e A358576   46: (o((o)(o)))
%e A358576   47: (((o)((o))))
%e A358576   49: ((oo)(oo))
%e A358576   51: ((o)((oo)))
%e A358576   57: ((o)(ooo))
%e A358576   60: (oo(o)((o)))
%e A358576   61: ((o(o)(o)))
%t A358576 MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A358576 Select[Range[100],Count[MGTree[#],_[__],{0,Infinity}]==Depth[MGTree[#]]-1&]
%Y A358576 The version for edge-height is A209638.
%Y A358576 Square trees are A358577, counted by A358589, ordered A358590.
%Y A358576 The version for leaves instead of height is A358578, counted by A185650.
%Y A358576 These trees are counted by A358587, ordered A358588.
%Y A358576 A000081 counts rooted trees, ordered A000108.
%Y A358576 A034781 counts rooted trees by nodes and height.
%Y A358576 A055277 counts rooted trees by leaves, ordered A001263.
%Y A358576 MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
%Y A358576 Cf. A000040, A000720, A001222, A007097, A056239, A112798.
%Y A358576 Cf. A206487, A358580, A358581-A358586, A358592, A358729.
%K A358576 nonn
%O A358576 1,1
%A A358576 _Gus Wiseman_, Nov 25 2022