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A214567 Maximal number of distinct rooted trees obtained from the rooted tree with Matula-Goebel number n by adding one pendant edge at one of its vertices. 2
1, 2, 3, 2, 4, 4, 3, 2, 3, 5, 5, 4, 5, 4, 6, 2, 4, 4, 3, 5, 5, 6, 4, 4, 4, 6, 3, 4, 6, 7, 6, 2, 7, 5, 6, 4, 5, 4, 7, 5, 6, 6, 5, 6, 6, 5, 7, 4, 3, 5, 6, 6, 3, 4, 8, 4, 5, 7, 5, 7, 5, 7, 5, 2, 8, 8, 4, 5, 6, 7, 6, 4, 6, 6, 6, 4, 7, 8, 7, 5, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The Matula-Goebel number of a rooted tree can be 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.

Sum_{j such that rooted tree with Matula-Goebel number j has n vertices} a(j) = A000107(n). Example: the Matula-Goebel numbers of the rooted trees with 4 vertices are 5,6,7,8 and a(5)+a(6)+a(7)+a(8) = 4+4+3+2=13 = A000107(4).

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 Y-N. 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, Rooted tree statistics from Matula numbers, arXiv:1111.4288.

Index entries for sequences related to Matula-Goebel numbers

FORMULA

a(1)=1; if n is t-th prime, then a(n)=1+a(t); if n is composite, then a(n) = 1+Sum_{p|n}(a(p)-1), where summation is over the distinct prime divisors of n.

EXAMPLE

a(4)=2 because the rooted tree with Matula-Goebel number 4 is V; adding an edge at either of the two leaves yields the same rooted tree.

a(5)=4 because the rooted tree with Matula-Goebel number 5 is the path on 4 vertices; adding one edge at any of the vertices yields a new rooted tree.

a(987654321)=18 (reader may verify this on Fig. 2 of the Deutsch paper).

MAPLE

with(numtheory): a := proc (n) local FS: FS := proc (n) options operator, arrow: factorset(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+a(pi(n)) else 1+add(a(FS(n)[j])-1, j = 1 .. nops(FS(n))) end if end proc: seq(a(n), n = 1 .. 130);

PROG

(Haskell)

import Data.List (genericIndex)

a214567 n = genericIndex a214567_list (n - 1)

a214567_list = 1 : g 2 where

  g x = y : g (x + 1) where

    y | t > 0     = a214567 t + 1

      | otherwise = 1 + sum (map ((subtract 1) . a214567) $ a027748_row x)

       where t = a049084 x

-- Reinhard Zumkeller, Sep 03 2013

CROSSREFS

Cf. A000107.

Cf. A049084, a027748.

Sequence in context: A214906 A274228 A321476 * A259363 A087437 A304739

Adjacent sequences:  A214564 A214565 A214566 * A214568 A214569 A214570

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jul 25 2012

STATUS

approved

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Last modified January 20 16:38 EST 2019. Contains 319335 sequences. (Running on oeis4.)