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A198328 The Matula-Goebel number of the rooted tree obtained from the rooted tree with Matula-Goebel number n after removing the leaves, together with their incident edges. 2
1, 1, 2, 1, 3, 2, 2, 1, 4, 3, 5, 2, 3, 2, 6, 1, 3, 4, 2, 3, 4, 5, 7, 2, 9, 3, 8, 2, 5, 6, 11, 1, 10, 3, 6, 4, 3, 2, 6, 3, 5, 4, 3, 5, 12, 7, 13, 2, 4, 9, 6, 3, 2, 8, 15, 2, 4, 5, 5, 6, 7, 11, 8, 1, 9, 10, 3, 3, 14, 6, 5, 4, 7, 3, 18, 2, 10, 6, 11, 3, 16, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This is not the pruning operation mentioned in the Balaban reference (p. 360) and in the Todeschini-Consonni reference (p. 42) since in the case that the root has degree 1, this root and the incident edge are not deleted.

REFERENCES

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

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)=1; a(2)=1; if n=p(t) (the t-th prime, t>1), then a(n)=p(a(t)); if n=rs (r,s,>=2), then a(n)=a(r)a(s). The Maple program is based on this recursive formula.

Completely multiplicative with a(2) = 1, a(prime(t)) = prime(a(t)) for t > 1. - Andrew Howroyd, Aug 01 2018

EXAMPLE

a(7)=2 because the rooted tree with Matula-Goebel number 7 is Y; after deleting the leaves and their incident edges, we obtain the 1-edge tree having Matula-Goebel number 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 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(a(pi(n))) else a(r(n))*a(s(n)) end if end proc; seq(a(n), n = 1 .. 120);

MATHEMATICA

a[1] = a[2] = 1; a[n_] := a[n] = If[PrimeQ[n], Prime[a[PrimePi[n]]], Times @@ (a[#[[1]]]^#[[2]]& /@ FactorInteger[n])];

Array[a, 100] (* Jean-Fran├žois Alcover, Dec 18 2017 *)

PROG

(Haskell)

import Data.List (genericIndex)

a198328 n = genericIndex a198328_list (n - 1)

a198328_list = 1 : 1 : g 3 where

   g x = y : g (x + 1) where

     y = if t > 0 then a000040 (a198328 t) else a198328 r * a198328 s

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

-- Reinhard Zumkeller, Sep 03 2013

CROSSREFS

Cf. A198329.

Cf. A049084, A020639, A000040.

Sequence in context: A008687 A080801 A124758 * A227277 A071481 A318832

Adjacent sequences:  A198325 A198326 A198327 * A198329 A198330 A198331

KEYWORD

nonn,mult

AUTHOR

Emeric Deutsch, Nov 24 2011

STATUS

approved

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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)