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A196063 The Narumi-Katayama index of the rooted tree with Matula-Goebel number n. 2
0, 1, 2, 2, 4, 4, 3, 3, 8, 8, 8, 6, 6, 6, 16, 4, 6, 12, 4, 12, 12, 16, 12, 8, 32, 12, 24, 9, 12, 24, 16, 5, 32, 12, 24, 16, 8, 8, 24, 16, 12, 18, 9, 24, 48, 24, 24, 10, 18, 48, 24, 18, 5, 32, 64, 12, 16, 24, 12, 32, 16, 32, 36, 6, 48, 48, 8, 18, 48, 36, 16, 20, 18, 16, 96, 12, 48, 36, 24, 20, 64, 24, 24, 24, 48, 18, 48, 32, 10, 64 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The Narumi-Katayama index of a connected graph is the product of the degrees of the vertices of the graph.

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.

H. Narumi and M. Katayama, Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Engin. Hokkaido Univ., 16, 1984, 209-214.

Z. Tomovic and I. Gutman, Narumi-Katayama index of phenylenes, J. Serb. Chem. Soc., 66(4), 2001, 243-247.

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; a(2)=1; if n = p(t) (the t-th prime, t>=2), then a(n)=a(t)*(1+G(t))/G(t); if n=rs (r,s>=2), then a(n)=a(r)*a(s)*G(n)/[G(r)*G(s)]; G(m) denotes the number of prime divisors of m counted with multiplicities. The Maple program is based on this recursive formula.

EXAMPLE

a(7)=3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1*3*1*1=3).

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: 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 n = 2 then 1 elif bigomega(n) = 1 then a(pi(n))*(1+bigomega(pi(n)))/bigomega(pi(n)) else a(r(n))*a(s(n))*bigomega(n)/(bigomega(r(n))*bigomega(s(n))) end if end proc: seq(a(n), n = 1 .. 90);

PROG

(Haskell)

import Data.List (genericIndex)

a196063 n = genericIndex a196063_list (n - 1)

a196063_list = 0 : 1 : g 3 where

   g x = y : g (x + 1) where

     y | t > 0     = a196063 t * (a001222 t + 1) `div` a001222 t

       | otherwise = a196063 r * a196063 s * a001222 x `div`

                     (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, A196065.

Sequence in context: A263856 A090277 A024222 * A205450 A215674 A279211

Adjacent sequences:  A196060 A196061 A196062 * A196064 A196065 A196066

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 01 2011

STATUS

approved

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Last modified February 20 01:26 EST 2018. Contains 299357 sequences. (Running on oeis4.)