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A196046 Maximum vertex-degree in the rooted tree with Matula-Goebel number n. 3
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 (list; graph; refs; listen; history; text; internal format)
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.

REFERENCES

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 Review, 10, 1968, 273.

LINKS

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

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011

Index entries for sequences related to Matula-Goebel numbers

FORMULA

a(1)=0; if n=p_t (=the t-th prime), then a(n) = max(a(t), 1+G(t)); if n=rs (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);

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

Cf. A049084, A020639, A001222.

Sequence in context: A292138 A322665 A273632 * A152724 A081743 A247069

Adjacent sequences:  A196043 A196044 A196045 * A196047 A196048 A196049

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 26 2011

STATUS

approved

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Last modified January 20 23:20 EST 2019. Contains 319343 sequences. (Running on oeis4.)