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A196058
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Diameter (i.e. largest distance between two vertices) of the rooted tree with Matula-Goebel number n.
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1
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0, 1, 2, 2, 3, 3, 2, 2, 4, 4, 4, 3, 3, 3, 5, 2, 3, 4, 2, 4, 4, 5, 4, 3, 6, 4, 4, 3, 4, 5, 5, 2, 6, 4, 5, 4, 3, 3, 5, 4, 4, 4, 3, 5, 5, 4, 5, 3, 4, 6, 5, 4, 2, 4, 7, 3, 4, 5, 4, 5, 4, 6, 4, 2, 6, 6, 3, 4, 5, 5, 4, 4, 4, 4, 6, 3, 6, 5, 5, 4, 4, 5, 4, 4, 6, 4, 6, 5, 3, 5, 5, 4, 7, 5, 5, 3, 6, 4, 6, 6, 4, 5, 4, 4, 5, 3, 3, 4, 5, 7, 5, 3, 5, 4, 6, 5, 5, 5, 5, 5
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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E. Deutsch, Tree statistics from Matula numbers, Arxiv preprint arXiv:1111.4288, 2011
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.
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LINKS
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Table of n, a(n) for n=1..120.
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FORMULA
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a(1)=0; if n=p(t) (=the t-th prime), then a(n)=max(a(t), 1+H(t)); if n=rs (r,s,>=2), then a(n)=max(a(r), a(s), H(r)+H(s)), where H(m) is the height of the tree with Matula-Goebel number m (see A109082). The Maple program is based on this recursive formula.
The Gutman et al. references contain a different recursive formula.
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EXAMPLE
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a(2^m) = 2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
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MAPLE
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with(numtheory): a := proc (n) local r, s, H: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: H := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+H(pi(n)) else max(H(r(n)), H(s(n))) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then max(a(pi(n)), 1+H(pi(n))) else max(a(r(n)), a(s(n)), H(r(n))+H(s(n))) end if end proc: seq(a(n), n = 1 .. 120);
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CROSSREFS
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Cf. A109082.
Sequence in context: A076902 A049113 A055093 * A081844 A110012 A023514
Adjacent sequences: A196055 A196056 A196057 * A196059 A196060 A196061
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Sep 30 2011
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STATUS
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approved
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