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A191517 Maximum edge-degree in the rooted tree with Matula-Goebel number n. 0
0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 4, 2, 3, 3, 4, 3, 3, 2, 4, 2, 3, 3, 4, 4, 4, 3, 4, 3, 4, 4, 3, 3, 3, 3, 5, 3, 3, 3, 4, 4, 4, 2, 5, 4, 3, 3, 4, 4, 2, 4, 5, 3, 3, 4, 3, 3, 4, 4, 5, 4, 4, 3, 5, 3, 4, 3, 5, 4, 3, 3, 5, 3, 4, 3, 4, 5, 4, 3, 4, 2, 3, 4, 6, 3, 4, 3, 4, 4, 3, 4, 5, 4, 5, 5, 5, 3, 3, 4, 6, 4, 5, 3, 4, 4, 3, 3, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,4

COMMENTS

The degree of an edge is the number of edges adjacent to it.

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.

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

LINKS

Table of n, a(n) for n=2..120.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288

Index entries for sequences related to Matula-Goebel numbers

FORMULA

In A191516 one finds the generating polynomial f(n)=F(n,x) of the edges of the rooted tree with Matula-Goebel number n, with respect to edge-degree. a(n)=degree of this polynomial.

EXAMPLE

a(7)=2 because the rooted tree with Matula-Goebel number 7 is Y; all edges have degree 2.

MAPLE

with(numtheory): f := proc (n) local r, s, g, h: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: g := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then x^bigomega(pi(n)) else x^bigomega(s(n))*g(r(n))+x^bigomega(r(n))*g(s(n)) end if end proc: h := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then x*g(pi(n))+h(pi(n)) else h(r(n))+h(s(n)) end if end proc: sort(expand(g(n)+h(n))) end proc: seq(degree(f(n)), n = 2 .. 120);

CROSSREFS

A191516

Sequence in context: A177227 A174373 A232270 * A303370 A082528 A055980

Adjacent sequences:  A191514 A191515 A191516 * A191518 A191519 A191520

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 15 2011

STATUS

approved

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Last modified December 5 23:38 EST 2021. Contains 349558 sequences. (Running on oeis4.)