A196046 - OEIS

A196046

Maximum vertex-degree in the rooted tree with Matula-Goebel number n.

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

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OFFSET

1,3

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.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

FORMULA

a(1)=0; if n=prime(t) (=the t-th prime), then a(n) = max(a(t), 1+G(t)); if n=r*s (r,s>=2), then a(n)=max(a(r),a(s), G(r)+G(s)); G(m) is the number of prime divisors of m counted with multiplicity. The Maple program is based on this recursive formula.

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

EXAMPLE

a(7)=3 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 bigomega(n) = 1 then max(a(pi(n)), 1+bigomega(pi(n))) else max(a(r(n)), a(s(n)), bigomega(r(n))+bigomega(s(n))) end if end proc: seq(a(n), n = 1 .. 110);

MATHEMATICA

r[n_] := FactorInteger[n][[1, 1]];

s[n_] := n/r[n];

a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, Max[a[PrimePi[n]], 1 + PrimeOmega[PrimePi[n]]], True, Max[a[r[n]], a[s[n]], PrimeOmega[r[n]] + PrimeOmega[s[n]]]];

Table[a[n], {n, 1, 110}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)

PROG

(Haskell)

import Data.List (genericIndex)

a196046 n = genericIndex a196046_list (n - 1)

a196046_list = 0 : g 2 where

g x = y : g (x + 1) where

y | t > 0 = max (a196046 t) (a001222 t + 1)

| otherwise = maximum [a196046 r, a196046 s, a001222 r + a001222 s]

where t = a049084 x; r = a020639 x; s = x `div` r

-- Reinhard Zumkeller, Sep 03 2013

CROSSREFS