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 A184169 Maximum escape distance over the vertices of the rooted tree having Matula-Goebel number n. 0
 0, 1, 2, 1, 3, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 3, 3, 1, 3, 1, 2, 1, 2, 2, 5, 1, 3, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 3, 2, 2, 3, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 2, 4, 2, 2, 4, 2, 1, 2, 3, 3, 2, 2, 2, 2, 1, 3, 1, 2, 1, 3, 1, 3, 2, 2, 2, 4, 1, 3, 1, 2, 3, 2, 2, 2, 2, 4, 2, 2, 1, 4, 1, 3, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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 E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011 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 degree of the polynomial P. EXAMPLE a(7)=2 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. 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: degree(P(n)) end proc: seq(a(n), n = 1 .. 110); CROSSREFS Sequence in context: A343073 A184166 A029423 * A176207 A059130 A094959 Adjacent sequences:  A184166 A184167 A184168 * A184170 A184171 A184172 KEYWORD nonn AUTHOR Emeric Deutsch, Oct 23 2011 STATUS approved

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Last modified June 23 17:39 EDT 2021. Contains 345402 sequences. (Running on oeis4.)