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

1,1

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.

Links

Table of n, a(n) for n=1..35.

Example

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))))))

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[#]]&]

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.

Sequence in context: A277572 A107622 A049306 * A023635 A317649 A265036

Adjacent sequences: A358728 A358729 A358730 * A358732 A358733 A358734

Keyword

nonn,more

Author

Gus Wiseman, Dec 01 2022

Status

approved