A109082 Depth of rooted tree having Matula-Goebel number n.

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

Offset: 1

Author

Keith Briggs, Aug 17 2005

Keywords

• nonn

Comments

Another term for depth is height.

Starting with n, a(n) is the number of times one must take the product of prime indices (A003963) to reach 1. - Gus Wiseman, Mar 27 2019

Examples

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

Links

• François Marques, Table of n, a(n) for n = 1..10000 (terms 1..5381 from Alois P. Heinz).
• Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 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.
• D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

A left inverse of A007097.

Cf. A003963, A061775, A091233.

Cf. A000081, A000720, A001222, A109129, A112798, A196050, A290822, A317713, A320325, A324927 (positions of 2), A324928 (positions of 3), A325032.

This statistic is counted by A034781, ordered A080936.

The ordered version is A358379.

For node-height instead of edge-height we have A358552.

Cf. A000108, A055277, A056239, A342507, A358576.

Programs

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-François Alcover, May 06 2014, after Emeric Deutsch *)

PARI

a(n) = my(v=factor(n)[,1],d=0); while(#v,d++; v=fold(setunion, apply(p->factor(primepi(p))[,1]~, v))); d; \\ Kevin Ryde, Sep 21 2020

Python

from functools import lru_cache
from sympy import isprime, primepi, primefactors
@lru_cache(maxsize=None)
def A109082(n):
    if n == 1 : return 0
    if isprime(n): return 1+A109082(primepi(n))
    return max(A109082(p) for p in primefactors(n)) # Chai Wah Wu, Mar 19 2022

Formula

a(1)=0; if n is the t-th prime, then a(n) = 1 + a(t); if n is composite, n=t*s, then a(n) = max(a(t),a(s)). The Maple program is based on this.

a(A007097(n)) = n.

a(n) = A358552(n) - 1. - Gus Wiseman, Nov 27 2022

Extensions

Edited by Emeric Deutsch, Sep 16 2011