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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
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.
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-François 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.
Sequence in context: A365391 A336390 A124758 * A366790 A369459 A369467
KEYWORD
nonn,mult
AUTHOR
Emeric Deutsch, Nov 24 2011
STATUS
approved

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Last modified March 19 02:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)