OFFSET
1,2
COMMENTS
The vertices of a rooted tree can be regarded as a partially ordered set, where u<=v holds for two vertices u and v if and only if u lies on the unique path between v and the root. A chain is a nonempty set of pairwise comparable vertices.
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
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
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 Rev. 10 (1968) 273.
FORMULA
a(1)=1; if n=prime(t), then a(n)=1+2a(t); if n=r*s (r,s,>=2), then a(n)=a(r)+a(s)-1. The Maple program is based on this recursive formula.
EXAMPLE
a(5) = 15 because the rooted tree with Matula-Goebel number 5 is a path ABCD on 4 vertices and any nonempty subset of {A,B,C,D} is a chain.
MAPLE
with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+2*a(pi(n)) else a(r(n))+a(s(n))-1 end if end proc: seq(a(n), n = 1 .. 80);
MATHEMATICA
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
a[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + 2*a[PrimePi[n]], True, a[r[n]] + a[s[n]] - 1];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 24 2024, after Maple code *)
PROG
(Haskell)
import Data.List (genericIndex)
a184162 n = genericIndex a184162_list (n - 1)
a184162_list = 1 : g 2 where
g x = y : g (x + 1) where
y = if t > 0 then 2 * a184162 t + 1 else a184162 r + a184162 s - 1
where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013
(PARI) a(n) = my(f=factor(n)); [self()(primepi(p)) |p<-f[, 1]] * f[, 2]*2 + 1; \\ Kevin Ryde, Aug 25 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 19 2011
STATUS
approved