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

Author

Gus Wiseman, Nov 27 2022

Keywords

• nonn
• more

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.

Examples

			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))))
		

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.

Programs

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]}]