A358554 Least Matula-Goebel number of a rooted tree with n internal (non-leaf) nodes.

1, 2, 3, 5, 11, 25, 55, 121, 275, 605, 1331, 3025, 6655, 14641, 33275, 73205

Offset

1,2

Comments

Positions of first appearances in A342507.

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..16.

Example

The terms together with their corresponding rooted trees begin:
      1: o
      2: (o)
      3: ((o))
      5: (((o)))
     11: ((((o))))
     25: (((o))((o)))
     55: (((o))(((o))))
    121: ((((o)))(((o))))
    275: (((o))((o))(((o))))
    605: (((o))(((o)))(((o))))
   1331: ((((o)))(((o)))(((o))))
   3025: (((o))((o))(((o)))(((o))))
   6655: (((o))(((o)))(((o)))(((o))))
  14641: ((((o)))(((o)))(((o)))(((o))))
  33275: (((o))((o))(((o)))(((o)))(((o))))
  73205: (((o))(((o)))(((o)))(((o)))(((o))))

Mathematica

MGTree[n_]:=If[n==1, {}, MGTree/@Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
seq=Table[Count[MGTree[n], _[__], {0, Infinity}], {n, 1000}];
Table[Position[seq, n][[1, 1]], {n, Union[seq]}]

Crossrefs

For height instead of internals we have A007097, firsts of A109082.

For leaves instead of internals we have A151821, firsts of A109129.

Positions of first appearances in A342507.

The ordered version gives firsts of A358553.

A000081 counts rooted trees, ordered A000108.

A034781 counts rooted trees by nodes and height.

A055277 counts rooted trees by nodes and leaves.

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

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

Cf. A001263, A206487, A358578, A358592.

Sequence in context: A060696 A076051 A000628 * A369495 A273755 A258804

Adjacent sequences: A358551 A358552 A358553 * A358555 A358556 A358557

Keyword

nonn,more

Author

Gus Wiseman, Nov 27 2022

Status

approved