login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A198329 The Matula-Goebel number of the rooted tree obtained from the rooted tree with Matula-Goebel number n after removing the vertices of degree one, together with their incident edges. 5
1, 1, 1, 1, 2, 2, 1, 1, 4, 3, 3, 2, 2, 2, 6, 1, 2, 4, 1, 3, 4, 5, 4, 2, 9, 3, 8, 2, 3, 6, 5, 1, 10, 3, 6, 4, 2, 2, 6, 3, 3, 4, 2, 5, 12, 7, 6, 2, 4, 9, 6, 3, 1, 8, 15, 2, 4, 5, 3, 6, 4, 11, 8, 1, 9, 10, 2, 3, 14, 6, 3, 4, 4, 3, 18, 2, 10, 6, 5, 3, 16, 5, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

This is the pruning operation mentioned, for example, in the Balaban reference (p. 360) and in the Todeschini - Consonni reference (p. 42).

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

Table of n, a(n) for n=1..83.

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

Let b(n)=A198328(n) (= the Matula-Goebel number of the tree obtained by removing the leaves together with their incident edges from the rooted tree with Matula-Goebel number n). a(1)=1; if n = p(t) (=the t-th prime), then a(n)=b(t); if n=rs (r,s>=2), then a(n)=b(r)b(s). The Maple program is based on this recursive formula.

EXAMPLE

a(7)=1 because the rooted tree with Matula-Goebel number 7 is Y; after deleting the vertices of degree one and their incident edges we obtain the 1-vertex tree having Matula-Goebel number 1.

MAPLE

with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: seq(a(n), n = 1 .. 120);

MATHEMATICA

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

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

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

CROSSREFS

Cf. A198328.

Sequence in context: A202735 A116595 A128315 * A123566 A279863 A201757

Adjacent sequences:  A198326 A198327 A198328 * A198330 A198331 A198332

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 24 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 23 03:08 EST 2019. Contains 319370 sequences. (Running on oeis4.)