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.
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 30 2011
STATUS
approved