

A196051


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


6



0, 1, 4, 4, 10, 10, 9, 9, 20, 20, 20, 18, 18, 18, 35, 16, 18, 31, 16, 32, 32, 35, 31, 28, 56, 31, 48, 29, 32, 50, 35, 25, 56, 32, 52, 44, 28, 28, 50, 46, 31, 46, 29, 52, 72, 48, 50, 40, 48, 75, 52, 46, 25, 64, 84, 42, 46, 50, 32, 67, 44, 56, 67, 36, 76, 76, 28, 48, 72, 70, 46, 59, 46, 44, 102, 42, 79, 68, 52, 62, 88, 50, 48, 62, 79, 46, 75, 71, 40, 92, 71, 67, 84, 72, 71, 54, 75, 65, 104, 96
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OFFSET

1,3


COMMENTS

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.
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.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=0; if n = p(t) (the tth prime), then a(n)=a(t)+PL(t)+E(t)+1; if n=rs (r,s>=2), then a(n)=a(r)+a(s)+PL(r)E(s)+PL(s)E(r); PL(m) and E(m) denote the path length and the number of edges of the rooted tree with Matula number m (see A196047, A196050). The Maple program is based on this recursive formula.


EXAMPLE

a(7)=9 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y (1+1+1+2+2+2=9).
a(2^m) = m^2 because the rooted tree with MatulaGoebel number 2^m is a star with m edges and we have m distances 1 and m(m1)/2 distances 2; m + m(m1)=m^2.


MAPLE

with(numtheory): a := proc (n) local r, s, E, PL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: E := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(r(n))+E(s(n)) end if end proc: PL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n))+PL(pi(n)) else PL(r(n))+PL(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n))+PL(pi(n))+1+E(pi(n)) else a(r(n))+a(s(n))+PL(r(n))*E(s(n))+PL(s(n))*E(r(n)) end if end proc: seq(a(n), n = 1 .. 100);


PROG

(Haskell)
import Data.List (genericIndex)
a196051 n = genericIndex a196051_list (n  1)
a196051_list = 0 : g 2 where
g x = y : g (x + 1) where
y  t > 0 = a196051 t + a196047 t + a196050 t + 1
 otherwise = a196051 r + a196051 s +
a196047 r * a196050 s + a196047 s * a196050 r
where t = a049084 x; r = a020639 x; s = x `div` r
 Reinhard Zumkeller, Sep 03 2013


CROSSREFS

Cf. A196047, A196050.
Cf. A049084, A020639.
Sequence in context: A050348 A134637 A078910 * A140234 A220044 A219828
Adjacent sequences: A196048 A196049 A196050 * A196052 A196053 A196054


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Sep 27 2011


STATUS

approved



