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A109129 Width (i.e., number of non-root vertices having degree 1) of the rooted tree with Matula-Goebel number n. 38
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

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

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.

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

Cf. A061775, A091233.

Cf. A049084, A020639.

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

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

Sequence in context: A052304 A049874 A060501 * A304486 A188550 A064122

Adjacent sequences:  A109126 A109127 A109128 * A109130 A109131 A109132

KEYWORD

nonn

AUTHOR

Keith Briggs, Aug 17 2005

EXTENSIONS

Typo in formula fixed by Reinhard Zumkeller, Sep 03 2013

STATUS

approved

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Last modified May 26 13:49 EDT 2020. Contains 334626 sequences. (Running on oeis4.)