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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. 5

%I #16 Dec 18 2017 08:30:09

%S 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 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 2,3,14,6,3,4,4,3,18,2,10,6,5,3,16,5,7

%N 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 This is the pruning operation mentioned, for example, in the Balaban reference (p. 360) and in the Todeschini - Consonni reference (p. 42).

%D A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.

%D R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.

%H 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 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 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 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 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F 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 = p(t) (=the t-th prime), then a(n)=b(t); if n=rs (r,s>=2), then a(n)=b(r)b(s). The Maple program is based on this recursive formula.

%e 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 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 b[1] = b[2] = 1; b[n_] := b[n] = If[PrimeQ[n], Prime[b[PrimePi[n]]], Times @@ (b[#[[1]]]^#[[2]]& /@ FactorInteger[n])];

%t a[1] = 1; a[n_] := a[n] = If[PrimeQ[n], b[PrimePi[n]], Times @@ (b[#[[1]] ]^#[[2]]& /@ FactorInteger[n])];

%t Array[a, 100] (* _Jean-François Alcover_, Dec 18 2017 *)

%Y Cf. A198328.

%K nonn

%O 1,5

%A _Emeric Deutsch_, Nov 24 2011

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