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A358577 Matula-Goebel numbers of "square" rooted trees, i.e., whose height equals their number of leaves.

%I A358577 #10 Nov 25 2022 16:41:39
%S A358577 1,4,12,14,18,19,21,27,40,52,60,68,70,74,78,86,89,90,91,92,95,100,102,
%T A358577 105,107,111,117,119,122,129,130,134,135,138,146,150,151,153,161,163,
%U A358577 169,170,175,176,181,183,185,195,201,206,207,215,219,221,225,227,230
%N A358577 Matula-Goebel numbers of "square" rooted trees, i.e., whose height equals their number of leaves.
%C A358577 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 A358577 Gus Wiseman, <a href="/A358577/a358577.png">The first 64 square rooted trees.</a>
%F A358577 A358552(a(n)) = A109129(a(n)).
%e A358577 The terms together with their corresponding rooted trees begin:
%e A358577    1: o
%e A358577    4: (oo)
%e A358577   12: (oo(o))
%e A358577   14: (o(oo))
%e A358577   18: (o(o)(o))
%e A358577   19: ((ooo))
%e A358577   21: ((o)(oo))
%e A358577   27: ((o)(o)(o))
%e A358577   40: (ooo((o)))
%e A358577   52: (oo(o(o)))
%e A358577   60: (oo(o)((o)))
%e A358577   68: (oo((oo)))
%e A358577   70: (o((o))(oo))
%e A358577   74: (o(oo(o)))
%e A358577   78: (o(o)(o(o)))
%e A358577   86: (o(o(oo)))
%e A358577   89: ((ooo(o)))
%e A358577   90: (o(o)(o)((o)))
%t A358577 MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A358577 Select[Range[100],Count[MGTree[#],{},{0,Infinity}]==Depth[MGTree[#]]-1&]
%Y A358577 Internals instead of leaves: A358576, counted by A358587, ordered A358588.
%Y A358577 Internals instead of height: A358578, counted by A185650, ordered A358579.
%Y A358577 These trees are counted by A358589, ordered A358590.
%Y A358577 A000081 counts rooted trees, ordered A000108.
%Y A358577 A034781 counts trees by nodes and height.
%Y A358577 A055277 counts trees by nodes and leaves, ordered A001263.
%Y A358577 MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
%Y A358577 Cf. A000040, A000720, A001222, A007097, A056239, A112798.
%Y A358577 Cf. A206487, A358379, A358580, A358581-A358586, A358592.
%K A358577 nonn
%O A358577 1,2
%A A358577 _Gus Wiseman_, Nov 25 2022