

A196058


Diameter (i.e., largest distance between two vertices) of the rooted tree with MatulaGoebel 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
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OFFSET

1,3


COMMENTS

The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel 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 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=0; if n=p(t) (=the tth 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 MatulaGoebel 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 MatulaGoebel 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}] (* JeanFrançois 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



