A196068 Visitation length of the rooted tree with Matula-Goebel number n.

1, 3, 6, 5, 10, 8, 9, 7, 11, 12, 15, 10, 13, 11, 15, 9, 14, 13, 12, 14, 14, 17, 17, 12, 19, 15, 16, 13, 18, 17, 21, 11, 20, 16, 18, 15, 16, 14, 18, 16, 19, 16, 17, 19, 20, 19, 22, 14, 17, 21, 19, 17, 15, 18, 24, 15, 17, 20, 20, 19, 20, 23, 19, 13, 22, 22, 18, 18, 22, 20, 21, 17, 21, 18, 24, 16, 23, 20, 24, 18

Offset: 1

Author

Emeric Deutsch, Oct 04 2011

Keywords

• nonn

Comments

The visitation length of a rooted tree is defined as the sum of the path length and the number of vertices. The path length of a rooted tree is defined as the sum of distances of all vertices to the root of the tree (see the Keijzer et al. reference).

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.

Examples

			a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1+2+2+4=9).
a(2^m) = 2m+1 because the rooted tree with Matula-Goebel number 2^m is a star with m edges (m+(m+1)=2m+1).
		

References

• 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.
• M. Keijzer and J. Foster, Crossover bias in genetic programming, Lecture Notes in Computer Sciences, 4445, 2007, 33-44.
• 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
• Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A196046, A196047.

Cf. A049084, A020639, A061775.

Programs

Haskell

import Data.List (genericIndex)
a196068 n = genericIndex a196068_list (n - 1)
a196068_list = 1 : g 2 where
   g x = y : g (x + 1) where
     y | t > 0     = a196068 t + a061775 t + 1
       | otherwise = a196068 r + a196068 s - 1
       where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013

Maple

with(numtheory): a := proc (n) local r, s, N: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: N := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then 1+N(pi(n)) else N(r(n))+N(s(n))-1 end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n))+N(pi(n))+1 else a(r(n))+a(s(n))-1 end if end proc: seq(a(n), n = 1 .. 80);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
nn[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + nn[PrimePi[n]], True, nn[r[n]] + nn[s[n]] - 1];
a[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, a[PrimePi[n]] + nn[PrimePi[n]] + 1, True, a[r[n]] + a[s[n]] - 1];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)

Formula

a(1)=1; if n=prime(t) (= the t-th prime) then a(n)=a(t)+N(t)+1, where N(t) is the number of nodes of the rooted tree with Matula number t; if n=r*s (r,s>=2), then a(n)=a(r)+a(s)-1. The Maple program is based on this recursive formula.