A109129 Width (i.e., number of non-root vertices having degree 1) of the rooted tree with Matula-Goebel number n.

0, 1, 1, 2, 1, 2, 2, 3, 2, 2, 1, 3, 2, 3, 2, 4, 2, 3, 3, 3, 3, 2, 2, 4, 2, 3, 3, 4, 2, 3, 1, 5, 2, 3, 3, 4, 3, 4, 3, 4, 2, 4, 3, 3, 3, 3, 2, 5, 4, 3, 3, 4, 4, 4, 2, 5, 4, 3, 2, 4, 3, 2, 4, 6, 3, 3, 3, 4, 3, 4, 3, 5, 3, 4, 3, 5, 3, 4, 2, 5, 4, 3, 2, 5, 3, 4, 3, 4, 4, 4, 4, 4, 2, 3, 4, 6, 2, 5, 3, 4

Offset: 1

Author

Keith Briggs, Aug 17 2005

Keywords

• nonn

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.

A non-root vertex having degree 1 is called a leaf.

Every positive integer has a unique factorization (see A324924) into factors q(i) = prime(i)/i for i > 0. The number of ones in this factorization is a(n). For example, 30 = q(1)^3 q(2)^2 q(3), so a(30) = 3. - Gus Wiseman, Mar 23 2019

Examples

			a(7)=2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• E. 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. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A061775, A091233.

Cf. A049084, A020639.

Cf. A000081, A000720, A001222, A007097, A109082, A196050, A317713.

Cf. A324850, A324922, A324923, A324924, A324931.

Programs

Haskell

import Data.List (genericIndex)
a109129 n = genericIndex a109129_list (n - 1)
a109129_list = 0 : 1 : g 3 where
   g x = y : g (x + 1) where
     y = if t > 0 then a109129 t else a109129 r + a109129 s
         where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013

Maple

with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);

Mathematica

Nest[Function[{a, n}, Append[a, If[PrimeQ@ n, a[[PrimePi@ n]], Total@ Map[#2 a[[#1]] & @@ # &, FactorInteger[n]] ]]] @@ {#, Length@ # + 1} &, {0, 1}, 105] (* Michael De Vlieger, Mar 24 2019 *)

PARI

ML(n) = if( n==1, 1, my(f=factor(n)); sum(k=1,matsize(f)[1],ML(primepi(f[k,1]))*f[k,2])) ;
A109129(n) = if( n==1, 0, ML(n) ); \\ François Marques, Mar 16 2021

Python

from functools import lru_cache
from sympy import primepi, isprime, factorint
@lru_cache(maxsize=None)
def A109129(n):
    if n <= 2: return n-1
    if isprime(n): return A109129(primepi(n))
    return sum(e*A109129(p) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022

Formula

a(1)=0; a(2)=1; if n = p(t) (= the t-th prime) and t >= 2, then a(n) = a(t); if n = rs (r, s >= 2), then a(n) = a(r) + a(s). The Maple program is based on this recursive formula.

The Gutman et al. references contain a different recursive formula.

Extensions

Typo in formula fixed by Reinhard Zumkeller, Sep 03 2013