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 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.
REFERENCES
A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
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.
R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.
LINKS
FORMULA
a(1)=0; if n=prime(t) (the t-th prime, t>=2), then a(n)=a(t)+2G(t); if n=r*s (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 Matula-Goebel 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);
MATHEMATICA
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, a[PrimePi[n]] + 2*PrimeOmega[ PrimePi[n]], True, a[r[n]]+a[s[n]]+2*PrimeOmega[r[n]]*PrimeOmega[s[n]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Nov 25 2011
STATUS
approved