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A224458 The Gordon-Scantlebury index of the rooted tree with Matula-Goebel number n. 2
0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 6, 4, 5, 6, 5, 5, 4, 5, 7, 5, 5, 6, 6, 5, 6, 4, 10, 5, 5, 6, 8, 7, 7, 6, 8, 5, 7, 6, 6, 7, 6, 6, 11, 7, 7, 6, 7, 10, 9, 6, 9, 8, 6, 5, 9, 8, 5, 8, 15, 7, 7, 7, 7, 7, 8, 8, 12, 7, 8, 8, 9, 7, 8, 6, 12, 10, 6, 6, 10, 7, 7, 7, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

The Gordon-Scantlebury index of a tree is the number of paths of length 2 between distinct vertices of the tree. See the Trinajstic reference (p. 115). It is 1/2 of the Platt index of the tree (A198332).

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.

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.

N. Trinajstic, Chemical Graph Theory, Vol. II, CRC Press, Boca Raton, 1983.

E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.

LINKS

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

Index entries for sequences related to Matula-Goebel numbers

FORMULA

a(1)=0; if n=p(t) (the t-th prime, t>=2), then a(n)=a(t)+G(t); if n=rs (r,s>=2), then a(n)=a(r)+a(s)+G(r)G(s); G(m) denotes the number of prime divisors of m counted with multiplicities.

EXAMPLE

a(7)=3 because the rooted tree with Matula-Goebel number 7 is Y; obviously, it has 3 paths of length 2.

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 a(pi(n))+bigomega(pi(n)) else a(r(n))+a(s(n))+bigomega(r(n))*bigomega(s(n)) end if end proc: seq(a(n), n = 1 .. 100);

PROG

(Haskell)

import Data.List (genericIndex)

a224458 n = genericIndex a224458_list (n - 1)

a224458_list = 0 : g 2 where

   g x = y : g (x + 1) where

     y | t > 0     = a224458 t + a001222 t

       | otherwise = a224458 r + a224458 s + a001222 r * a001222 s

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

-- Reinhard Zumkeller, Sep 03 2013

CROSSREFS

Cf. A198332.

Cf. A049084, A020639, A001222.

Sequence in context: A111393 A062537 A279596 * A097688 A262685 A171895

Adjacent sequences:  A224455 A224456 A224457 * A224459 A224460 A224461

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Apr 14 2013

STATUS

approved

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Last modified January 16 15:53 EST 2019. Contains 319195 sequences. (Running on oeis4.)