

A109129


Width (i.e., number of nonroot vertices having degree 1) of the rooted tree with MatulaGoebel number n.


73



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
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OFFSET

1,4


COMMENTS

The MatulaGoebel number of a rooted tree is defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.
A nonroot 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 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=0; a(2)=1; if n = p(t) (= the tth 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 MatulaGoebel number 7 is the rooted tree Y.
a(2^m) = m because the rooted tree with MatulaGoebel 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
(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 n1
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


CROSSREFS

Cf. A061775, A091233.
Cf. A049084, A020639.
Cf. A000081, A000720, A001222, A007097, A109082, A196050, A317713.
Cf. A324850, A324922, A324923, A324924, A324931.
Sequence in context: A049874 A060501 A355661 * 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



