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A109082 Depth of rooted tree having Matula-Goebel number n. 16
0, 1, 2, 1, 3, 2, 2, 1, 2, 3, 4, 2, 3, 2, 3, 1, 3, 2, 2, 3, 2, 4, 3, 2, 3, 3, 2, 2, 4, 3, 5, 1, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 4, 3, 3, 4, 2, 2, 3, 3, 3, 2, 2, 4, 2, 2, 4, 4, 3, 3, 5, 2, 1, 3, 4, 3, 3, 3, 3, 4, 2, 3, 3, 3, 2, 4, 3, 5, 3, 2, 4, 4, 2, 3, 3, 4, 4, 3, 3, 3, 3, 5, 4, 3, 2, 4, 2, 4, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Another term for depth is height.

REFERENCES

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..5381

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011

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.

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=ts, then a(n) = max(a(t),a(s)). The Maple program is based on this.

a(A007097(n)) = n.

EXAMPLE

a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the 3-edge rooted tree Y of height 2.

MAPLE

with(numtheory): a := proc(n) option remember; if n = 1 then 0 elif isprime(n) then 1+a(pi(n)) else max((map (p->a(p), factorset(n)))[]) end if end proc: seq(a(n), n = 1 .. 100); # Emeric Deutsch, Sep 16 2011

MATHEMATICA

a [n_] := a[n] = If[n == 1, 0, If[PrimeQ[n], 1+a[PrimePi[n]], Max[Map[a, FactorInteger[n][[All, 1]]]]]]; Table[a[n], {n, 1, 100}] (* Jean-Fran├žois Alcover, May 06 2014, after Emeric Deutsch *)

CROSSREFS

A left inverse of A007097.

Cf. A003963, A061775, A091233.

Sequence in context: A096857 A303639 A090000 * A126303 A157810 A072339

Adjacent sequences:  A109079 A109080 A109081 * A109083 A109084 A109085

KEYWORD

nonn

AUTHOR

Keith Briggs (keith.briggs(AT)bt.com), Aug 17 2005

EXTENSIONS

Edited by Emeric Deutsch, Sep 16 2011

STATUS

approved

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Last modified January 16 23:44 EST 2019. Contains 319206 sequences. (Running on oeis4.)