A358577 Matula-Goebel numbers of "square" rooted trees, i.e., whose height equals their number of leaves.

1, 4, 12, 14, 18, 19, 21, 27, 40, 52, 60, 68, 70, 74, 78, 86, 89, 90, 91, 92, 95, 100, 102, 105, 107, 111, 117, 119, 122, 129, 130, 134, 135, 138, 146, 150, 151, 153, 161, 163, 169, 170, 175, 176, 181, 183, 185, 195, 201, 206, 207, 215, 219, 221, 225, 227, 230

Offset

1,2

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 square rooted trees.

Formula

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

Example

The terms together with their corresponding rooted trees begin:
   1: o
   4: (oo)
  12: (oo(o))
  14: (o(oo))
  18: (o(o)(o))
  19: ((ooo))
  21: ((o)(oo))
  27: ((o)(o)(o))
  40: (ooo((o)))
  52: (oo(o(o)))
  60: (oo(o)((o)))
  68: (oo((oo)))
  70: (o((o))(oo))
  74: (o(oo(o)))
  78: (o(o)(o(o)))
  86: (o(o(oo)))
  89: ((ooo(o)))
  90: (o(o)(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}]==Depth[MGTree[#]]-1&]

Crossrefs

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

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

These trees are counted by A358589, ordered A358590.

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, A358379, A358580, A358581-A358586, A358592.

Sequence in context: A031330 A052339 A310562 * A123076 A310563 A260257

Adjacent sequences: A358574 A358575 A358576 * A358578 A358579 A358580

Keyword

nonn

Author

Gus Wiseman, Nov 25 2022

Status

approved