cp's OEIS Frontend

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.

%I A198329 #20 Jun 27 2024 10:19:49
%S A198329 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,
%T A198329 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,
%U A198329 2,3,14,6,3,4,4,3,18,2,10,6,5,3,16,5,7
%N 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.
%C A198329 This is the pruning operation mentioned, for example, in the Balaban reference (p. 360) and in the Todeschini - Consonni reference (p. 42).
%D A198329 A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
%D A198329 R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.
%H A198329 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 A198329 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 A198329 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 A198329 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 A198329 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A198329 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 = prime(t), then a(n)=b(t); if n=r*s (r,s>=2), then a(n)=b(r)*b(s). The Maple program is based on this recursive formula.
%e A198329 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.
%p A198329 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);
%t A198329 b[1] = b[2] = 1; b[n_] := b[n] = If[PrimeQ[n], Prime[b[PrimePi[n]]], Times @@ (b[#[[1]]]^#[[2]]& /@ FactorInteger[n])];
%t A198329 a[1] = 1; a[n_] := a[n] = If[PrimeQ[n], b[PrimePi[n]], Times @@ (b[#[[1]] ]^#[[2]]& /@ FactorInteger[n])];
%t A198329 Array[a, 100] (* _Jean-François Alcover_, Dec 18 2017 *)
%Y A198329 Cf. A198328.
%K A198329 nonn
%O A198329 1,5
%A A198329 _Emeric Deutsch_, Nov 24 2011