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A196048 External path length of the rooted tree with Matula-Goebel number n. 2
0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 4, 4, 5, 5, 5, 4, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 6, 6, 6, 5, 5, 6, 7, 7, 6, 7, 7, 7, 6, 7, 7, 8, 6, 7, 7, 7, 6, 8, 7, 8, 7, 8, 7, 7, 7, 8, 7, 8, 7, 8, 6, 8, 6, 8, 7, 9, 8, 8, 8, 8, 7, 9, 8, 8, 8, 8, 8, 7, 7, 8, 8, 8, 8, 9, 9, 8, 7, 9, 8, 9, 8, 7, 8, 9, 7, 8, 9, 8, 8, 9, 9, 9, 8, 9, 9, 10, 8, 8, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The external path length of a rooted tree is defined as the sum of the distances of all leaves to the root.

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

Table of n, a(n) for n=1..110.

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; t>1) then a(n)=a(t)+LV(t), where LV(t) is the number of leaves in the rooted tree with Matula number t; if n=rs (r,s>=2), then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.

EXAMPLE

a(7)=4 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (2+2=4).

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, LV: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LV := proc (n) if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then LV(pi(n)) else LV(r(n))+LV(s(n)) end if end proc: if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n))+LV(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);

PROG

(Haskell)

import Data.List (genericIndex)

a196048 n = genericIndex a196048_list (n - 1)

a196048_list = 0 : 1 : g 3 where

   g x = y : g (x + 1) where

     y = if t > 0 then a196048 t + a109129 t else a196048 r + a196048 s

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

-- Reinhard Zumkeller, Sep 03 2013

CROSSREFS

Cf. A109129, A196047.

Cf. A049084, A020639.

Sequence in context: A157790 A070241 A066412 * A176075 A256555 A117119

Adjacent sequences:  A196045 A196046 A196047 * A196049 A196050 A196051

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 27 2011

EXTENSIONS

Offset fixed by Reinhard Zumkeller, Sep 03 2013

STATUS

approved

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Last modified January 19 16:32 EST 2019. Contains 319309 sequences. (Running on oeis4.)