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 A198337 Radius of rooted tree having Matula-Goebel number n. 0

%I

%S 0,1,1,1,2,2,1,1,2,2,2,2,2,2,3,1,2,2,1,2,2,3,2,2,3,2,2,2,2,3,3,1,3,2,

%T 3,2,2,2,3,2,2,2,2,3,3,2,3,2,2,3,3,2,1,2,4,2,2,3,2,3,2,3,2,1,3,3,2,2,

%U 3,3,2,2,2,2,3,2,3,3,3,2,2,3,2,2,3,2

%N Radius of rooted tree having Matula-Goebel number n.

%C The radius of a tree is defined as the minimum eccentricity of the vertices.

%C The radius of a tree is equal to the number of prunings required to reduce the tree to the 1-vertex tree. See the Balaban reference, p. 360.

%C The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

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

%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

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

%F A198336(n) gives the sequence of the Matula-Goebel numbers of the rooted trees obtained from the rooted tree with Matula-Goebel number n by pruning it successively 0,1,2,... times. Then the radius of the rooted tree with Matula-Goebel number n is equal to the number of terms in this sequence diminished by 1.

%e a(7)=1 because the rooted tree with Matula-Goebel number 7 is Y and its vertices have eccentricities 2,2,2,1. a(11)=2 because the rooted tree with Matula-Goebel number 11 is the path tree on 5 vertices and the eccentricities are 4,4,3,3,2.

%p with(numtheory): aa := 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: S := proc (m) local A, i: A[m, 1] := m; for i while aa(A[m, i]) < A[m, i] do A[m, i+1] := aa(A[m, i]) end do: seq(A[m, j], j = 1 .. i) end proc; a := proc (n) options operator, arrow: nops([S(n)])-1 end proc: seq(a(n), n = 1 .. 110);

%Y Cf. A198336.

%K nonn

%O 1,5

%A _Emeric Deutsch_, Dec 01 2011

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Last modified December 8 23:37 EST 2021. Contains 349596 sequences. (Running on oeis4.)