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A198332 The Platt index of the rooted tree with Matula-Goebel number n. 3
0, 0, 2, 2, 4, 4, 6, 6, 6, 6, 6, 8, 8, 8, 8, 12, 8, 10, 12, 10, 10, 8, 10, 14, 10, 10, 12, 12, 10, 12, 8, 20, 10, 10, 12, 16, 14, 14, 12, 16, 10, 14, 12, 12, 14, 12, 12, 22, 14, 14, 12, 14, 20, 18, 12, 18, 16, 12, 10, 18, 16, 10, 16, 30, 14, 14, 14, 14, 14 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The Platt index (or Platt number or total edge adjacency index) of a tree is the sum of the degrees of all the edges (degree of an edge = number of edges adjacent to it). See the Todeschini-Consonni reference (p. 125). It is also equal to 2 x number of paths of length 2.

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.

A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.

R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.

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)+2G(t); if n=rs (r,s>=2), then a(n)=a(r)+a(s)+2G(r)G(s); G(m) denotes the number of prime di visors of m counted with multiplicities.

EXAMPLE

a(7)=6 because the rooted tree with Matula-Goebel number 7 is Y, where each edge has degree 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))+2*bigomega(pi(n)) else a(r(n))+a(s(n))+2*bigomega(r(n))*bigomega(s(n)) end if end proc: seq(a(n), n = 1 .. 90);

PROG

(Haskell)

import Data.List (genericIndex)

a198332 n = genericIndex a198332_list (n - 1)

a198332_list = 0 : g 2 where

   g x = y : g (x + 1) where

     y | t > 0     = a198332 t + 2 * a001222 t

       | otherwise = a198332 r + a198332 s + 2 * 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, A224458.

Sequence in context: A007730 A257686 A057144 * A080606 A083535 A250983

Adjacent sequences:  A198329 A198330 A198331 * A198333 A198334 A198335

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 25 2011

STATUS

approved

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Last modified January 18 16:40 EST 2019. Contains 319271 sequences. (Running on oeis4.)