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A196049 Number of branching nodes of the rooted tree with Matula-Goebel number n. 2
0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,28

COMMENTS

A branching node of a tree is a vertex of degree at least 3.

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) and t is not the product of 2 prime factors, then a(n)=a(t); if n=p(t) (= the t-th prime) and t is the product of 2 prime factors, then a(n)=a(t)+1; if n=rs (r prime, s>=2) and s is not a product of 2 prime factors, then a(n)=a(r)+a(s); if n=rs (r prime, s>=2) and s is a product of 2 prime factors, then a(n)=a(r)+a(s)+1. The Maple program is based on this recursive formula.

EXAMPLE

a(7)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.

if m>2 then a(2^m) = 1 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 and bigomega(pi(n)) <> 2 then a(pi(n)) elif bigomega(n) = 1 then a(pi(n))+1 elif bigomega(s(n)) <> 2 then a(r(n))+a(s(n)) else a(r(n))+a(s(n))+1 end if end proc: seq(a(n), n = 1 .. 110);

PROG

(Haskell)

import Data.List (genericIndex)

a196049 n = genericIndex a196049_list (n - 1)

a196049_list = 0 : g 2 where

   g x = y : g (x + 1) where

     y | t > 0     = a196049 t + a064911 t

       | otherwise = a196049 r + a196049 s + a064911 s

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

-- Reinhard Zumkeller, Sep 03 2013

CROSSREFS

Cf. A049084, A020639, A064911.

Sequence in context: A259287 A090464 A277967 * A194821 A044934 A124761

Adjacent sequences:  A196046 A196047 A196048 * A196050 A196051 A196052

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 27 2011

STATUS

approved

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Last modified January 20 02:05 EST 2019. Contains 319320 sequences. (Running on oeis4.)