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%I #11 Mar 07 2017 11:22:46
%S 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,
%T 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,
%U 7,7,8,8,12,7,8,8,9,7,8,6,12,10,6,6,10,7,7,7,9
%N The Gordon-Scantlebury index of the rooted tree with Matula-Goebel number n.
%C 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).
%C 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.
%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D N. Trinajstic, Chemical Graph Theory, Vol. II, CRC Press, Boca Raton, 1983.
%D E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
%H Reinhard Zumkeller, <a href="/A224458/b224458.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F 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.
%e a(7)=3 because the rooted tree with Matula-Goebel number 7 is Y; obviously, it has 3 paths of length 2.
%p 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);
%o (Haskell)
%o import Data.List (genericIndex)
%o a224458 n = genericIndex a224458_list (n - 1)
%o a224458_list = 0 : g 2 where
%o g x = y : g (x + 1) where
%o y | t > 0 = a224458 t + a001222 t
%o | otherwise = a224458 r + a224458 s + a001222 r * a001222 s
%o where t = a049084 x; r = a020639 x; s = x `div` r
%o -- _Reinhard Zumkeller_, Sep 03 2013
%Y Cf. A198332.
%Y Cf. A049084, A020639, A001222.
%K nonn
%O 1,5
%A _Emeric Deutsch_, Apr 14 2013
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