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A224458
The Gordon-Scantlebury index of the rooted tree with Matula-Goebel number n.
2
0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 6, 4, 5, 6, 5, 5, 4, 5, 7, 5, 5, 6, 6, 5, 6, 4, 10, 5, 5, 6, 8, 7, 7, 6, 8, 5, 7, 6, 6, 7, 6, 6, 11, 7, 7, 6, 7, 10, 9, 6, 9, 8, 6, 5, 9, 8, 5, 8, 15, 7, 7, 7, 7, 7, 8, 8, 12, 7, 8, 8, 9, 7, 8, 6, 12, 10, 6, 6, 10, 7, 7, 7, 9
OFFSET
1,5
COMMENTS
The Gordon-Scantlebury index of a tree is the number of paths of length 2 between distinct vertices of the tree. See the Trinajstic reference (p. 115). It is 1/2 of the Platt index of the tree (A198332).
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.
REFERENCES
Emeric Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
N. Trinajstic, Chemical Graph Theory, Vol. II, CRC Press, Boca Raton, 1983.
LINKS
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
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)=0; if n=prime(t) (the t-th prime, t>=2), then a(n)=a(t)+G(t); if n=r*s (r,s>=2), then a(n)=a(r)+a(s)+G(r)*G(s); G(m) denotes the number of prime divisors of m counted with multiplicities.
EXAMPLE
a(7)=3 because the rooted tree with Matula-Goebel number 7 is Y; obviously, it has 3 paths of length 2.
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 bigomega(n) = 1 then a(pi(n))+bigomega(pi(n)) else a(r(n))+a(s(n))+bigomega(r(n))*bigomega(s(n)) end if end proc: seq(a(n), n = 1 .. 100);
MATHEMATICA
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, a[PrimePi[n]] + PrimeOmega[ PrimePi[n]], True, a[r[n]]+a[s[n]] + PrimeOmega[r[n]]*PrimeOmega[s[n]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)
PROG
(Haskell)
import Data.List (genericIndex)
a224458 n = genericIndex a224458_list (n - 1)
a224458_list = 0 : g 2 where
g x = y : g (x + 1) where
y | t > 0 = a224458 t + a001222 t
| otherwise = a224458 r + a224458 s + 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, Apr 14 2013
STATUS
approved