A358727 Matula-Goebel numbers of rooted trees with greater number of leaves (width) than node-height.

8, 16, 24, 28, 32, 36, 38, 42, 48, 49, 53, 54, 56, 57, 63, 64, 72, 76, 80, 81, 84, 96, 98, 104, 106, 108, 112, 114, 120, 126, 128, 131, 133, 136, 140, 144, 147, 148, 152, 156, 159, 160, 162, 168, 171, 172, 178, 180, 182, 184, 189, 190, 192, 196, 200, 204, 208

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.

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

Links

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

Gus Wiseman, The first 64 rooted trees with greater width than height.

Example

The terms together with their corresponding rooted trees begin:
   8: (ooo)
  16: (oooo)
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  42: (o(o)(oo))
  48: (oooo(o))
  49: ((oo)(oo))
  53: ((oooo))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  57: ((o)(ooo))
  63: ((o)(o)(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  76: (oo(ooo))

Mathematica

MGTree[n_]:=If[n==1, {}, MGTree/@Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Depth[MGTree[#]]-1<Count[MGTree[#], {}, {-2}]&]

Crossrefs

Positions of negative terms in A358726.

These trees are counted by A358728.

Differences: A358580, A358724, A358726, A358729.

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.

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

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

Cf. A185650, A206487, A209638, A358576, A358577, A358578, A358587, A358730.

Sequence in context: A365866 A033859 A177899 * A246311 A344538 A244371

Adjacent sequences: A358724 A358725 A358726 * A358728 A358729 A358730

Keyword

nonn

Author

Gus Wiseman, Dec 01 2022

Status

approved