

A198332


The Platt index of the rooted tree with MatulaGoebel number n.


3



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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OFFSET

1,3


COMMENTS

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 TodeschiniConsonni reference (p. 125). It is also equal to 2 x number of paths of length 2.
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.
A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355375, 1979.
R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, WileyVCH, 2000.


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)+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.


EXAMPLE

a(7)=6 because the rooted tree with MatulaGoebel number 7 is Y, where each edge has degree 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))+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);


PROG

(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
 Reinhard Zumkeller, Sep 03 2013


CROSSREFS

Cf. A049084, A020639, A001222, A224458.
Sequence in context: A330271 A257686 A057144 * A080606 A083535 A250983
Adjacent sequences: A198329 A198330 A198331 * A198333 A198334 A198335


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Nov 25 2011


STATUS

approved



