OFFSET
1,3
COMMENTS
The Narumi-Katayama index of a connected graph is the product of the degrees of the vertices of the graph.
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
Emeric Deutsch, Tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
Emeric Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
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.
H. Narumi and M. Katayama, Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Engin. Hokkaido Univ., 16, 1984, 209-214.
Z. Tomovic and I. Gutman, Narumi-Katayama index of phenylenes, J. Serb. Chem. Soc., 66(4), 2001, 243-247.
FORMULA
a(1)=0; a(2)=1; if n = prime(t) (the t-th prime, t>=2), then a(n)=a(t)*(1+G(t))/G(t); if n=r*s (r,s>=2), then a(n)=a(r)*a(s)*G(n)/[G(r)*G(s)]; G(m) denotes the number of prime divisors of m counted with multiplicities. The Maple program is based on this recursive formula.
EXAMPLE
a(7)=3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1*3*1*1=3).
a(2^m) = m 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: 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 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n))*(1+bigomega(pi(n)))/bigomega(pi(n)) else a(r(n))*a(s(n))*bigomega(n)/(bigomega(r(n))*bigomega(s(n))) end if end proc: seq(a(n), n = 1 .. 90);
MATHEMATICA
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
a[n_] := Which[n == 1, 0, n == 2, 1, PrimeOmega[n] == 1, a[PrimePi[n]]*(1 + PrimeOmega[PrimePi[n]])/PrimeOmega[PrimePi[n]], True, a[r[n]]*a[s[n]]* PrimeOmega[n]/(PrimeOmega[r[n]]*PrimeOmega[s[n]])];
Table[a[n], {n, 1, 90}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)
PROG
(Haskell)
import Data.List (genericIndex)
a196063 n = genericIndex a196063_list (n - 1)
a196063_list = 0 : 1 : g 3 where
g x = y : g (x + 1) where
y | t > 0 = a196063 t * (a001222 t + 1) `div` a001222 t
| otherwise = a196063 r * a196063 s * a001222 x `div`
(a001222 r * a001222 s)
where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 01 2011
STATUS
approved