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A224458
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The Gordon-Scantlebury index of the rooted tree with Matula-Goebel number n.
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2
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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
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OFFSET
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1,5
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COMMENTS
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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.
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REFERENCES
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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 Review, 10, 1968, 273.
N. Trinajstic, Chemical Graph Theory, Vol. II, CRC Press, Boca Raton, 1983.
E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
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LINKS
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FORMULA
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a(1)=0; if n=p(t) (the t-th prime, t>=2), then a(n)=a(t)+G(t); if n=rs (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.
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EXAMPLE
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a(7)=3 because the rooted tree with Matula-Goebel number 7 is Y; obviously, it has 3 paths of length 2.
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MAPLE
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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);
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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