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A358592 Matula-Goebel numbers of rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.

%I A358592 #6 Nov 26 2022 12:35:20
%S A358592 18,21,60,70,78,91,92,95,102,111,119,122,129,146,151,181,201,227,264,
%T A358592 269,308,348,376,406,418,426,452,492,497,519,551,562,574,583,596,606,
%U A358592 659,664,668,698,707,708,717,779,794,796,809,826,834,911,932,934,942,958
%N A358592 Matula-Goebel numbers of rooted trees whose height, number of leaves, and number of internal (non-leaf) nodes are all equal.
%C A358592 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 A358592 Gus Wiseman, <a href="/A358592/a358592.png">The first 64 ordered trees whose height, number of leaves, and number of internal nodes are all equal.</a>
%F A358592 A358552(a(n)) = A342507(a(n)) = A109129(a(n)).
%e A358592 The terms together with their corresponding rooted trees begin:
%e A358592    18: (o(o)(o))
%e A358592    21: ((o)(oo))
%e A358592    60: (oo(o)((o)))
%e A358592    70: (o((o))(oo))
%e A358592    78: (o(o)(o(o)))
%e A358592    91: ((oo)(o(o)))
%e A358592    92: (oo((o)(o)))
%e A358592    95: (((o))(ooo))
%e A358592   102: (o(o)((oo)))
%e A358592   111: ((o)(oo(o)))
%e A358592   119: ((oo)((oo)))
%e A358592   122: (o(o(o)(o)))
%e A358592   129: ((o)(o(oo)))
%e A358592   146: (o((o)(oo)))
%e A358592   151: ((oo(o)(o)))
%e A358592   181: ((o(o)(oo)))
%e A358592   201: ((o)((ooo)))
%e A358592   227: (((oo)(oo)))
%t A358592 MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A358592 Select[Range[100],Count[MGTree[#],_[__],{0,Infinity}]==Count[MGTree[#],{},{0,Infinity}]==Depth[MGTree[#]]-1&]
%Y A358592 Any number of leaves: A358576, counted by A358587 (ordered A358588).
%Y A358592 Any number of internals: A358577, counted by A358589, ordered A358590.
%Y A358592 Any height: A358578, ordered A358579, counted by A185650.
%Y A358592 A000081 counts rooted trees, ordered A000108.
%Y A358592 A034781 counts rooted trees by nodes and height.
%Y A358592 A055277 counts rooted trees by nodes and leaves, ordered A001263.
%Y A358592 MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
%Y A358592 Cf. A000040, A000720, A001222, A007097, A056239, A112798.
%Y A358592 Cf. A206487, A358580, A358581-A358586.
%K A358592 nonn
%O A358592 1,1
%A A358592 _Gus Wiseman_, Nov 25 2022