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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.

%I A198328 #26 Jun 27 2024 10:19:56
%S A198328 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,
%T A198328 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,
%U A198328 10,3,3,14,6,5,4,7,3,18,2,10,6,11,3,16,5
%N 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.
%C A198328 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.
%D A198328 A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
%D A198328 R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.
%H A198328 Reinhard Zumkeller, <a href="/A198328/b198328.txt">Table of n, a(n) for n = 1..10000</a>
%H A198328 F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H A198328 I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A198328 I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H A198328 D. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H A198328 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A198328 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.
%F A198328 Completely multiplicative with a(2) = 1, a(prime(t)) = prime(a(t)) for t > 1. - _Andrew Howroyd_, Aug 01 2018
%e A198328 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.
%p A198328 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);
%t A198328 a[1] = a[2] = 1; a[n_] := a[n] = If[PrimeQ[n], Prime[a[PrimePi[n]]], Times @@ (a[#[[1]]]^#[[2]]& /@ FactorInteger[n])];
%t A198328 Array[a, 100] (* _Jean-François Alcover_, Dec 18 2017 *)
%o A198328 (Haskell)
%o A198328 import Data.List (genericIndex)
%o A198328 a198328 n = genericIndex a198328_list (n - 1)
%o A198328 a198328_list = 1 : 1 : g 3 where
%o A198328    g x = y : g (x + 1) where
%o A198328      y = if t > 0 then a000040 (a198328 t) else a198328 r * a198328 s
%o A198328          where t = a049084 x; r = a020639 x; s = x `div` r
%o A198328 -- _Reinhard Zumkeller_, Sep 03 2013
%Y A198328 Cf. A198329.
%Y A198328 Cf. A049084, A020639, A000040.
%K A198328 nonn,mult
%O A198328 1,3
%A A198328 _Emeric Deutsch_, Nov 24 2011