A196048 External path length of the rooted tree with Matula-Goebel number n.

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 4, 4, 5, 5, 5, 4, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 6, 6, 6, 5, 5, 6, 7, 7, 6, 7, 7, 7, 6, 7, 7, 8, 6, 7, 7, 7, 6, 8, 7, 8, 7, 8, 7, 7, 7, 8, 7, 8, 7, 8, 6, 8, 6, 8, 7, 9, 8, 8, 8, 8, 7, 9, 8, 8, 8, 8, 8, 7, 7, 8, 8, 8, 8, 9, 9, 8, 7, 9, 8, 9, 8, 7, 8, 9, 7, 8, 9, 8, 8, 9, 9, 9, 8, 9, 9, 10, 8, 8, 8

Offset: 1

Author

Emeric Deutsch, Sep 27 2011

Keywords

• nonn

Comments

The external path length of a rooted tree is defined as the sum of the distances of all leaves to the root.

The Matula-Goebel number of a rooted tree is 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)=4 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (2+2=4).
		

Links

• François Marques, Table of n, a(n) for n = 1..10000
• Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
• 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. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A109129, A196047.

Cf. A049084, A020639.

Programs

Haskell

import Data.List (genericIndex)
a196048 n = genericIndex a196048_list (n - 1)
a196048_list = 0 : 1 : g 3 where
   g x = y : g (x + 1) where
     y = if t > 0 then a196048 t + a109129 t else a196048 r + a196048 s
         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, LV: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LV := proc (n) if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then LV(pi(n)) else LV(r(n))+LV(s(n)) end if end proc: if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n))+LV(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);

Mathematica

a[m_] := Module[{r, s, LV},
   r[n_] := FactorInteger[n][[1, 1]];
   s[n_] := n/r[n];
   LV [n_] := Which[
      n == 1, 0,
      n == 2, 1,
      PrimeOmega[n] == 1, LV[PrimePi[n]],
      True, LV[r[n]] + LV[s[n]]];
   Which[
      m == 1, 0,
      m == 2, 1,
      PrimeOmega[m] == 1, a[PrimePi[m]] + LV[PrimePi[m]],
      True, a[r[m]] + a[s[m]]]];
Table[a[n], {n, 1, 110}] (* Jean-François Alcover, May 04 2023, after Maple code *)

PARI

LEpl(n) = { if(n==1, return([1,0]),
    my(f=factor(n)~, l, e, le);
      foreach(f,p,
        le=LEpl(primepi(p[1]));
        l+=le[1]*p[2];
        e+=(le[1]+le[2])*p[2];
      );
    return([l,e]) )
  };
A196048(n) = LEpl(n)[2]; \\ François Marques, Mar 14 2021

Formula

a(1)=0; a(2)=1; if n=prime(t) (the t-th prime; t>1) then a(n)=a(t)+LV(t), where LV(t) is the number of leaves in the rooted tree with Matula number t; if n=rs (r,s>=2), then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.

a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

Extensions

Offset fixed by Reinhard Zumkeller, Sep 03 2013