A358725 Matula-Goebel numbers of rooted trees with a greater number of internal (non-leaf) vertices than edge-height.

9, 15, 18, 21, 23, 25, 27, 30, 33, 35, 36, 39, 42, 45, 46, 47, 49, 50, 51, 54, 55, 57, 60, 61, 63, 65, 66, 69, 70, 72, 73, 75, 77, 78, 81, 83, 84, 85, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 102, 103, 105, 108, 110, 111, 113, 114, 115, 117, 119, 120, 121

Offset: 1

Author

Gus Wiseman, Nov 29 2022

Keywords

• nonn

Comments

Edge-height (A109082) is the number of edges in the longest path from root to leaf.

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.

Examples

			The terms together with their corresponding trees begin:
   9: ((o)(o))
  15: ((o)((o)))
  18: (o(o)(o))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  30: (o(o)((o)))
  33: ((o)(((o))))
  35: (((o))(oo))
  36: (oo(o)(o))
  39: ((o)(o(o)))
  42: (o(o)(oo))
  45: ((o)(o)((o)))
  46: (o((o)(o)))
  47: (((o)((o))))
  49: ((oo)(oo))
  50: (o((o))((o)))
		

Links

• Gus Wiseman, The first 64 ordered trees with a greater number of internal vertices than edge-height.

Crossrefs

Complement of A209638 (the case of equality).

These trees are counted by A316321.

Positions of positive terms in A358724.

The case of equality for node-height is A358576.

A000081 counts rooted trees, ordered A000108.

A034781 counts rooted trees by nodes and height, ordered A080936

A055277 counts rooted trees by nodes and leaves, ordered A001263.

Differences: A358580, A358724, A358726, A358729.

MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.

MG core: A000040, A000720, A001222, A007097, A056239, A112798.

Cf. A185650, A206487, A358577, A358578, A358581-A358586, A358587, A358592, A358730.

Programs

Mathematica

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],[_],{0,Infinity}]>Depth[MGTree[#]]-2&]

Formula

A342507(a(n)) > A109082(a(n)).