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A196058 Diameter (i.e., largest distance between two vertices) of the rooted tree with Matula-Goebel number n. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..120.

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.

F. Göbel, 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 Rev. 10 (1968) 273.

Index entries for sequences related to Matula-Goebel numbers

FORMULA

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.

EXAMPLE

a(2^m) = 2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

MAPLE

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);

MATHEMATICA

r[n_] := r[n] = FactorInteger[n][[1, 1]];

s[n_] := s[n] = n/r[n];

H[n_] := H[n] = Which[n == 1, 0, PrimeOmega[n] == 1, 1 + H[PrimePi[n]], True, Max[H[r[n]], H[s[n]]]];

a[n_] := a[n] = Which[n == 1, 0, PrimeOmega[n] == 1, Max[a[PrimePi[n]], 1 + H[PrimePi[n]]], True, Max[a[r[n]], a[s[n]], H[r[n]] + H[s[n]]]];

Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Nov 13 2017, after Emeric Deutsch *)

CROSSREFS

Cf. A109082.

Sequence in context: A290884 A049113 A055093 * A081844 A233549 A110012

Adjacent sequences:  A196055 A196056 A196057 * A196059 A196060 A196061

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 30 2011

STATUS

approved

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Last modified January 20 23:20 EST 2019. Contains 319343 sequences. (Running on oeis4.)