A358577 - OEIS

%I #10 Nov 25 2022 16:41:39

%S 1,4,12,14,18,19,21,27,40,52,60,68,70,74,78,86,89,90,91,92,95,100,102,

%T 105,107,111,117,119,122,129,130,134,135,138,146,150,151,153,161,163,

%U 169,170,175,176,181,183,185,195,201,206,207,215,219,221,225,227,230

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

%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="/A358577/a358577.png">The first 64 square rooted trees.</a>

%F A358552(a(n)) = A109129(a(n)).

%e The terms together with their corresponding rooted trees begin:

%e    1: o

%e    4: (oo)

%e   12: (oo(o))

%e   14: (o(oo))

%e   18: (o(o)(o))

%e   19: ((ooo))

%e   21: ((o)(oo))

%e   27: ((o)(o)(o))

%e   40: (ooo((o)))

%e   52: (oo(o(o)))

%e   60: (oo(o)((o)))

%e   68: (oo((oo)))

%e   70: (o((o))(oo))

%e   74: (o(oo(o)))

%e   78: (o(o)(o(o)))

%e   86: (o(o(oo)))

%e   89: ((ooo(o)))

%e   90: (o(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[#]]-1&]

%Y Internals instead of leaves: A358576, counted by A358587, ordered A358588.

%Y Internals instead of height: A358578, counted by A185650, ordered A358579.

%Y These trees are counted by A358589, ordered A358590.

%Y A000081 counts rooted trees, ordered A000108.

%Y A034781 counts trees by nodes and height.

%Y A055277 counts 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, A358379, A358580, A358581-A358586, A358592.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 25 2022