The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A184170 Number of vertices that have largest escape distance in the rooted tree having Matula-Goebel number n. 1
 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 3, 1, 3, 3, 2, 1, 4, 1, 1, 1, 1, 1, 3, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 1, 4, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS The escape distance of a vertex v in a rooted tree T is the distance from v to the nearest leaf of T that is a descendant of v. For the rooted tree ARBCDEF, rooted at R, the escape distance of B is 4 (the leaf A is not a descendant of B). 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 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. LINKS Table of n, a(n) for n=1..110. E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011 Index entries for sequences related to Matula-Goebel numbers FORMULA In A184167 one can find the generating polynomial P(n)=P(n,x) of the vertices of the rooted tree having Matula-Goebel number n, according to escape distance. a(n) is equal to the coefficient of the highest power of x. EXAMPLE a(7)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 4 vertices with escape distances 0,0,1, and 2; the largest value (2) occurs only once. a(15)=2 because the rooted tree with Matula number 15 is the path tree ABRCDE, rooted at R; the escape distances are 0,1,2,2,1,0; the largest value (2) occurs twice. MAPLE with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))-x^max(LLL(r(n)), LLL(s(n))) end if end proc: a := proc (n) options operator, arrow: coeff(P(n), x, degree(P(n))) end proc: seq(a(n), n = 1 .. 110); CROSSREFS Sequence in context: A326620 A353372 A305501 * A025919 A095684 A205565 Adjacent sequences: A184167 A184168 A184169 * A184171 A184172 A184173 KEYWORD nonn AUTHOR Emeric Deutsch, Oct 23 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 20 15:18 EDT 2024. Contains 371844 sequences. (Running on oeis4.)