A198332 - OEIS

A198332

The Platt index of the rooted tree with Matula-Goebel number n.

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

(list; graph; refs; listen; history; text; internal format)

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for sequences related to Matula-Goebel numbers

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