

A198337


Radius of rooted tree having MatulaGoebel number n.


0



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, 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, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 3, 2, 2, 3, 2
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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 1vertex tree. See the Balaban reference, p. 360.
The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


REFERENCES

A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355375, 1979.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.


LINKS

Table of n, a(n) for n=1..86.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

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


EXAMPLE

a(7)=1 because the rooted tree with MatulaGoebel number 7 is Y and its vertices have eccentricities 2,2,2,1. a(11)=2 because the rooted tree with MatulaGoebel 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);


CROSSREFS

Cf. A198336.
Sequence in context: A345699 A212632 A025885 * A206483 A087011 A294602
Adjacent sequences: A198334 A198335 A198336 * A198338 A198339 A198340


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Dec 01 2011


STATUS

approved



