A196058 Diameter (i.e., largest distance between two vertices) of the rooted tree with Matula-Goebel number n.

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

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

François Marques, Table of n, a(n) for n = 1..10000.

Emeric 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=prime(t), then a(n)=max(a(t), 1+H(t)); if n=r*s (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.

a(n^k) = 2*A109082(n) for k > 1. - François Marques, Mar 13 2021

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 *)

Prog

(PARI) HD(n) = { if(n==1, return([0, 0]),
           my(f=factor(n)~, h=0, d=0, hd);
           foreach(f, p,
             hd=HD(primepi(p[1]));
             hd[1]++;
             d=max(max(d, if(p[2]>1, 2*hd[1], hd[2])), h+hd[1]);
             h=max(h, hd[1])
           );
           return([h, d])
           )
};
A196058(n)=HD(n)[2]; \\ François Marques, Mar 13 2021

Crossrefs

Cf. A109082.

Sequence in context: A332249 A049113 A055093 * A081844 A233549 A334475

Adjacent sequences: A196055 A196056 A196057 * A196059 A196060 A196061

Keyword

nonn

Author

Emeric Deutsch, Sep 30 2011

Status

approved