A358731 Matula-Goebel numbers of rooted trees whose number of nodes is one more than their node-height.

4, 6, 7, 10, 13, 17, 22, 29, 41, 59, 62, 79, 109, 179, 254, 277, 293, 401, 599, 1063, 1418, 1609, 1787, 1913, 2749, 4397, 8527, 10762, 11827, 13613, 15299, 16519, 24859, 42043, 87803

Offset: 1

Author

Gus Wiseman, Dec 01 2022

Keywords

• nonn
• more

Comments

These are paths with a single extra leaf growing from them.

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.

Node-height is the number of nodes in the longest path from root to leaf.

Examples

			The terms together with their corresponding rooted trees begin:
    4: (oo)
    6: (o(o))
    7: ((oo))
   10: (o((o)))
   13: ((o(o)))
   17: (((oo)))
   22: (o(((o))))
   29: ((o((o))))
   41: (((o(o))))
   59: ((((oo))))
   62: (o((((o)))))
   79: ((o(((o)))))
  109: (((o((o)))))
  179: ((((o(o)))))
  254: (o(((((o))))))
  277: (((((oo)))))
  293: ((o((((o))))))
  401: (((o(((o))))))
  599: ((((o((o))))))
		

Crossrefs

These trees are counted by A289207.

Positions of 1's in A358729.

A000081 counts rooted trees, ordered A000108.

A034781 counts rooted trees by nodes and height.

A055277 counts rooted trees by nodes and leaves.

MG differences: A358580, A358724, A358726, A358729.

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

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

Cf. A206487, A209638, A316321, A358576, A358577, A358592, A358725.

Programs

Mathematica

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Count[MGTree[#],_,{0,Infinity}]==Depth[MGTree[#]]&]