

A198328


The MatulaGoebel number of the rooted tree obtained from the rooted tree with MatulaGoebel 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
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OFFSET

1,3


COMMENTS

This is not the pruning operation mentioned in the Balaban reference (p. 360) and in the TodeschiniConsonni 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, 355375, 1979.
R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, WileyVCH, 2000.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=1; a(2)=1; if n=p(t) (the tth 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 MatulaGoebel number 7 is Y; after deleting the leaves and their incident edges, we obtain the 1edge tree having MatulaGoebel 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] (* JeanFrançois 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 A324293
Adjacent sequences: A198325 A198326 A198327 * A198329 A198330 A198331


KEYWORD

nonn,mult


AUTHOR

Emeric Deutsch, Nov 24 2011


STATUS

approved



