A184166 The level of the lowest leaf of the rooted tree with Matula-Goebel number n.

0, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1

Offset

1,3

Comments

The Matula-Goebel number of a rooted tree is defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

References

F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

Links

Table of n, a(n) for n=1..110.

Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.

Index entries for sequences related to Matula-Goebel numbers

Formula

a(1)=0; if n is the t-th prime, then a(n) = 1 + a(t); if n is composite, n=rs, then a(n) = min(a(r),a(s)). The Maple program is based on this recursive rule.

Example

a(7)=2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (all leaves are at level 2).

Maple

with (numtheory):
a:= proc(n) option remember;
      if n=1 then 0
    elif isprime(n)=true then 1 +a(pi(n))
    else min (seq (a(i), i=factorset(n)))
      fi
    end:
seq (a(n), n=1..200);

Mathematica

a[n_] := a[n] = Which[
   n == 1, 0,
   PrimeQ[n], 1 + a[PrimePi[n]],
   True, Min[a /@ FactorInteger[n][[All, 1]]]];
Table[a[n], {n, 1, 200}] (* Jean-François Alcover, Jun 24 2024, after Maple code *)

Crossrefs

Cf. A184167.

Sequence in context: A376305 A214180 A343073 * A029423 A184169 A176207

Adjacent sequences: A184163 A184164 A184165 * A184167 A184168 A184169

Keyword

nonn

Author

Emeric Deutsch, Oct 22 2011

Status

approved