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A198332
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The Platt index of the rooted tree with Matula-Goebel number n.
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3
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0, 0, 2, 2, 4, 4, 6, 6, 6, 6, 6, 8, 8, 8, 8, 12, 8, 10, 12, 10, 10, 8, 10, 14, 10, 10, 12, 12, 10, 12, 8, 20, 10, 10, 12, 16, 14, 14, 12, 16, 10, 14, 12, 12, 14, 12, 12, 22, 14, 14, 12, 14, 20, 18, 12, 18, 16, 12, 10, 18, 16, 10, 16, 30, 14, 14, 14, 14, 14
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internal format)
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OFFSET
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1,3
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COMMENTS
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The Platt index (or Platt number or total edge adjacency index) of a tree is the sum of the degrees of all the edges (degree of an edge = number of edges adjacent to it). See the Todeschini-Consonni reference (p. 125). It is also equal to 2 x number of paths of length 2.
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.
A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.
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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)+2G(t); if n=rs (r,s>=2), then a(n)=a(r)+a(s)+2G(r)G(s); G(m) denotes the number of prime di visors of m counted with multiplicities.
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EXAMPLE
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a(7)=6 because the rooted tree with Matula-Goebel number 7 is Y, where each edge has degree 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))+2*bigomega(pi(n)) else a(r(n))+a(s(n))+2*bigomega(r(n))*bigomega(s(n)) end if end proc: seq(a(n), n = 1 .. 90);
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PROG
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(Haskell)
import Data.List (genericIndex)
a198332 n = genericIndex a198332_list (n - 1)
a198332_list = 0 : g 2 where
g x = y : g (x + 1) where
y | t > 0 = a198332 t + 2 * a001222 t
| otherwise = a198332 r + a198332 s + 2 * 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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