OFFSET
1,5
COMMENTS
The radius of a tree is defined as the minimum eccentricity of the vertices.
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.
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.
REFERENCES
A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
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. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
FORMULA
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.
EXAMPLE
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.
MAPLE
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);
MATHEMATICA
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
b[n_] := Which[n == 1, 1, n == 2, 1, PrimeOmega[n] == 1, Prime[b[PrimePi[n]]], True, b[r[n]]*b[s[n]]];
aa[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, b[PrimePi[n]], True, b[r[n]]*b[s[n]]];
S[m_] := Module[{A, i}, A[m, 1] = m; For[i = 1, aa[A[m, i]] < A[m, i], i++, A[m, i + 1] = aa[A[m, i]]]; Table[A[m, j], {j, 1, i}]];
a[n_] := Length[S[n]] - 1;
Table[a[n], {n, 1, 110}] (* Jean-François Alcover, Aug 12 2024, after Emeric Deutsch *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 01 2011
STATUS
approved