A358578 Matula-Goebel numbers of rooted trees whose number of leaves equals their number of internal (non-leaf) nodes.

2, 6, 7, 18, 20, 21, 26, 34, 37, 43, 54, 60, 63, 67, 70, 78, 88, 91, 92, 95, 102, 111, 116, 119, 122, 129, 142, 146, 151, 162, 164, 173, 180, 181, 189, 200, 201, 202, 210, 227, 234, 236, 239, 245, 260, 264, 269, 273, 276, 278, 285, 306, 308, 314, 322, 333, 337

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..57.

Gus Wiseman, The first 64 rooted trees whose number of leaves equals their number of internal nodes.

Formula

A342507(a(n)) = A109129(a(n)).

Example

The terms together with their corresponding rooted trees begin:
   2: (o)
   6: (o(o))
   7: ((oo))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  26: (o(o(o)))
  34: (o((oo)))
  37: ((oo(o)))
  43: ((o(oo)))
  54: (o(o)(o)(o))
  60: (oo(o)((o)))
  63: ((o)(o)(oo))
  67: (((ooo)))
  70: (o((o))(oo))
  78: (o(o)(o(o)))
  88: (ooo(((o))))
  91: ((oo)(o(o)))

Mathematica

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

Crossrefs

These trees are counted by A185650, ordered A358579.

Height instead of leaves: A358576, counted by A358587, ordered A358588.

Height instead of internals: A358577, counted by A358589, ordered A358590.

Positions of 0's in A358580.

A000081 counts rooted trees, ordered A000108.

A034781 counts trees by nodes and height.

A055277 counts trees by nodes and leaves, ordered A001263.

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

Cf. A000040, A000720, A001222, A007097, A056239, A112798.

Cf. A206487, A358371, A358581-A358586, A358592.

Sequence in context: A060133 A107784 A210619 * A095036 A290379 A342534

Adjacent sequences: A358575 A358576 A358577 * A358579 A358580 A358581

Keyword

nonn

Author

Gus Wiseman, Nov 25 2022

Status

approved