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.
FORMULA
a(1)=1; a(2)=1; if n=prime(t) (t>1), then a(n)=prime(a(t)); if n=r*s (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
KEYWORD
nonn,mult
AUTHOR
Emeric Deutsch, Nov 24 2011
STATUS
approved