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