

A224458


The GordonScantlebury index of the rooted tree with MatulaGoebel 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
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OFFSET

1,5


COMMENTS

The GordonScantlebury 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 MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


REFERENCES

F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
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, 23142322.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=0; if n=p(t) (the tth 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.


EXAMPLE

a(7)=3 because the rooted tree with MatulaGoebel 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);


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

Cf. A198332.
Cf. A049084, A020639, A001222.
Sequence in context: A323665 A062537 A279596 * A097688 A262685 A171895
Adjacent sequences: A224455 A224456 A224457 * A224459 A224460 A224461


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Apr 14 2013


STATUS

approved



