login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A196063 The Narumi-Katayama index of the rooted tree with Matula-Goebel number n. 2
0, 1, 2, 2, 4, 4, 3, 3, 8, 8, 8, 6, 6, 6, 16, 4, 6, 12, 4, 12, 12, 16, 12, 8, 32, 12, 24, 9, 12, 24, 16, 5, 32, 12, 24, 16, 8, 8, 24, 16, 12, 18, 9, 24, 48, 24, 24, 10, 18, 48, 24, 18, 5, 32, 64, 12, 16, 24, 12, 32, 16, 32, 36, 6, 48, 48, 8, 18, 48, 36, 16, 20, 18, 16, 96, 12, 48, 36, 24, 20, 64, 24, 24, 24, 48, 18, 48, 32, 10, 64 (list; graph; refs; listen; history; text; internal format)
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
E. Deutsch, Tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
E. 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.
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=rs (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);
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
Sequence in context: A090277 A324662 A024222 * A205450 A215674 A279211
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 01 2011
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 07:04 EDT 2024. Contains 370953 sequences. (Running on oeis4.)